acceptance samplng

8/22/2019 Acceptance Samplng

1/47

Acceptance Sampling


2/47

CustomerSupplier100%

Insp.

100%

Insp.

CustomerSupplier100%

Insp.

Sample

Insp.

CustomerSupplierSample

Insp.

SPC.


3/47

Content

Introductionand Basics of

acceptance

sampling

Part 1

Application ofStandard

Acceptance

sampling plans

Part 2


4/47

A procedure for sentencing incoming batches or lots of

items without doing 100% inspection

What is acceptance sampling?


5/47

Purpose of acceptance sampling?

Determine the average quality level of an incoming

shipment or at the end of production and judge whether

quality level is within the level that has been predetermined.


6/47

Why not 100% inspection?

Very expensive

Cant use when product must be destroyed to test

Handling by inspectors can itself induce defects

Inspection becomes tedious in order to prevent defective

items from slipping through inspection


7/47

Acceptance Sampling forAttributes


8/47

The general approach

N(Lot)

n

CountNumber

Conforming

Accept orReject Lot

Specify the sampling plan

For a lot size N, determine

the sample size (s) n, and

Select acceptance criteria for good lots

Select rejection criteria for bad lots

Accept the lot if acceptance criteria are satisfied

Specify course of action if lot is rejected


9/47

What's a good and bad lot ?

Acceptance quality level (AQL)

The smallest percentage of defectives that will make the

lot definitely acceptable. A quality level that is the base

line requirement of the customer

RQL or Lot tolerance percent defective (LTPD)

Quality level that is unacceptable to the customer


10/47

d Ac?

Reject lot

Yes

Accept lot

Do 100%

inspection

Return lot

to supplier

Inspect all items in the

sample

Defectives found = d

No

Take a

randomized

sample of size n

from the lot N

Example : Single

Sampling procedure


11/47

Sampling plans are based on sample statistics and the

theory says that since we inspect only a sample and not

the whole lot, there is a chance of making an error.


12/47

What is the probability of a head

appearing if you toss a balanced

and unbiased coin ?

A coin was tossed 10 times. The

number of times heads appeared was

recorded. This was repeated again

many times. The results are as follows.

2 3 6 4 5 3 8 9


13/47

This means, given an overall probability of

occurrence of an event, a sample (which is smaller

than the population) will have its own probability of

experiencing that event.


14/47

Example

100 Lots of size 5000 and known quality level of 1%

defectives were taken.

One sample of size = 100 was drawn from each lot.

If it was decided that if sample contained > 1 defective

units, it should be rejected.

In this case the acceptance number is 1


15/47

Simulation Example

Each square is a lot of size 5000. Sampling will be done using

a sample size of 100 and number of defectives will berecorded


16/47

0 3 0 0 1 2 2 0 1 1

1 0 0 0 2 2 0 0 0 1

1 1 0 1 0 0 1 0 0 0

1 2 0 3 1 0 0 0 3 0

1 1 0 0 1 1 0 0 2 2

1 0 0 0 0 2 1 0 0 2

2 1 1 0 1 0 0 1 2 1

0 0 0 1 0 2 1 2 1 0

1 1 1 3 0 0 4 0 2 2

0 1 1 1 1 0 0 0 3 1

Simulation Example

This is an output from a simulator which is set to

produce lots with 1 % defectives

We are supposed to reject lots where the sample contained

more than 1 defectives


17/47

0 3 0 0 1 2 2 0 1 1

1 0 0 0 2 2 0 0 0 1

1 1 0 1 0 0 1 0 0 0

1 2 0 3 1 0 0 0 3 0

1 1 0 0 1 1 0 0 2 2

1 0 0 0 0 2 1 0 0 2

2 1 1 0 1 0 0 1 2 1

0 0 0 1 0 2 1 2 1 0

1 1 1 3 0 0 4 0 2 2

0 1 1 1 1 0 0 0 3 1

Simulation Example

No. of lots rejected = 21

We are supposed to reject lots where the sample contained

more than 1 defectives


18/47

We need to consider two types of errors that result in

wrong decisions

Type 1 Error No Error

No Error Type 2 Error

Reject Accept

Good lot

Bad lot

T

R

U

T

H

DECISION


19/47

TYPE I ERROR = P (reject good lot)

orProducers risk5% is common

TYPE II ERROR = P (accept bad lot)

orConsumers risk10% is typical value

Errors and Risks


20/47

Producers & Consumers Risks

Producers risk

Risk associated with rejecting a lot of acceptable quality

Alpha () risk= Prob (committing Type I error)

= Prob (rejecting lot at AQL quality level)


21/47

Consumers risk

Receive shipment, assume good quality, actually

bad quality

Beta () risk= Prob (committing Type II error)

= Prob (accepting a lot at RQL quality level)

Producers & Consumers Risks


22/47

Probability of finding exactly x number of non

conforming units in a sample of size n, given that

proportion non conforming is p and the lot size is N, can

be determined using Binomial and Poisson distributions.


23/47

( )

D N D

x n xp x

N

n

D = Number of non conforming items in the population

N = Size of the population

n = sample size

X = number of non conforming items in the sample

Hypergeometric Distribution

Sampling from a finite population (lot) without replacement


24/47

P(x = c) = probability of exactly c non conforming units.

p = proportion non conforming

n = sample size

x = number of non conforming items

Binomial Distribution

cncpp

c

ncxP

)1()(

Sampling from a lot which is much greater than the sample

size n without replacement


25/47

Binomial Distribution

cncpp

cnc

ncxP

)1()!(!

!)(

P(x = c) = probability of exactly c non conforming units.

p = proportion non conforming

n = sample sizex = number of non conforming items


26/47

c

x

xnx ppxnx

ncxP0

)1()!(!

!)(

Probability of x c number of non conforming units in a

sample of size n, given that proportion non conforming

is p

Where has this formula come from ?


27/47

c

x

xnx

a

c

x

xnx

a

ppxnx

nP

pp

xnx

nP

0

22

0

11

)1()!(!

!

)1(

)!(!

!1

Probability of accepting a lot based on a sampling plan

with acceptance number as c for a AQL of p1 and a RQL

of p2

Where have these formulae come from ?


28/47

where,

= average number of nonconformities = np

Poisson Distribution

( ) !

xe

p xx

Probability of finding exactly x number of non

conforming units in a sample of size n, when theaverage number of nonconformities is some constant,

np, is:


29/47

A process is operating at a nonconformance level

of 1%. What is the probability that a sample of size

100 will have 2 defective units ?

p = 0.01, n = 100, x = 2

= np = 1

( )!

xe

p xx

1 2

(2)2!

ep

= 0.367


30/47

A process is operating at a nonconformance level

of 1%. What is the probability that a sample of size

100 will have 2 or less defective units ?

p = 0.01, n = 100, x = 2

= np = 1

( )!

xe

p xx

1 0 1 1 1 2(1) (1) (1)( 2)

0! 2! 2!

e e ep x

( 2) ( 0) ( 1) ( 2)p x p x p x p x


31/47

Binomial

Distribution

Poisson

Distribution ()

HypergeometricDistribution

n large,

p < 0.1

np < 5

Approximations

n / N < 0.1


32/47

0.5 1.0 1.5 2.0

0 0.6070 0.3680 0.2230 0.135

1 0.9100 0.7360 0.5580 0.406

2 0.9860 0.9200 0.8090 0.677

3 0.9980 0.9810 0.9340 0.857

4 1.0000 0.9960 0.9810 0.947

Ac np

From the Poisson distribution table above,

P (x 2 | np = 1) = 0.92 = Pa

np = 100 (0.01) = 1

Cumulative Poisson distribution table

A process is operating at a nonconformance level of 1%. What is

the probability that a sample of size 100 will have 2 or less

defective units ?


33/47

Rectification

InspectionPlans


34/47

Average outgoing quality

The average quality level of a series of lots that leave the

inspection station, assuming rectification, after coming

in for inspection at a certain quality level p.


35/47

What is rectification ?

A sample is selected from an incoming lot of quality p.

If number of non-conforming items are less than or equal

to Ac, the lot is accepted.

The non-conforming items are replaced with good ones.

If number of non-conforming items are greater than Ac,the lot is isolated and 100% inspection is carried out. All

the non-conforming items are replaced by good ones.

AOQ measures the average quality level of large number

of such batches of incoming quality p.

It is calculated by using the following formula.

( )aP N n pAOQN


36/47

Example

What is the AOQ for the following sampling plan ?

N = 2000, n = 50, Ac = 2, AQL = 2 % = 0.02

0.5 1.0 1.5 2.00 0.6070 0.3680 0.2230 0.135

1 0.9100 0.7360 0.5580 0.406

2 0.9860 0.9200 0.8090 0.677

3 0.9980 0.9810 0.9340 0.857

4 1.0000 0.9960 0.9810 0.947

Acnp

AOQ = [ Pa. p (N-n)] / N

AOQ = [ 0.92 x 0.02 x (2000 - 50)] / 2000

AOQ = 0.0179 = 1.79 %

np = 50 x 0.02 = 1


37/47

Average outgoing quality curve

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.05 0.1 0.15 0.2

Incomong quality (p)

AOQ


38/47

Average outgoing quality Limit (AOQL)

It is the peak of the AOQ curve.

It represents the worst average quality that wouldleave the inspection station, assuming rectification,

regardless of incoming quality


39/47

Average outgoing quality limit

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.05 0.1 0.15 0.2


AOQ

AOQL

It is the peak of the AOQ curve.


40/47

How much do I need to inspect ?

ATI


41/47

Average Total Inspection (ATI)

It is the average number of items inspected per lot

if rectifying inspection is conducted.

ATI can be used to calculate the average cost of

inspection.


42/47

ATI = n + (1- Pa) (N-n)

Average Total Inspection Curve

0

100

200

300

400

500

600

700

800

0.00 0.05 0.10 0.15 0.20


ATI

N=1000

N=25

D=2


43/47

Double Sampling Plans


44/47

Double Sampling Plans

n1 , n2 , Ac1 , Ac2 , Re1 , Re2

Take the first sample of size n1.

If the no. of defectives are Ac1, accept the lot.If the no of defectives are Re1, reject the lot.If the no. of defectives are > Ac1 and < Re1, take second sample

of size n2.

If the total no. of defectives ofboth the samples put together are Ac2, accept the lot.If the total no. of defectives ofboth the samples put together are Re2, reject the lot.


45/47

n1 = 25, n2 = 50, Ac1 = 2, Ac2 = 4, Re1 = 5, Re2 = 5

Probability of acceptance

For a given value of p,

Pa1 = P(x1 Ac1) = P(x1 2)

Pa2 = [ P(x1 = 3) P(x2 1) + P(x1 = 4) P(x2 = 0) ]

Overall probability of acceptance,

Pa = Pa1 + Pa2


46/47

Pa = 0.95 Pa = 0.5 Pa = 0.1

11.90 0 1 0.21 1.00 2.50

7.54 1 2 0.52 1.82 3.92

6.79 0 2 0.43 1.42 2.96

5.39 1 3 0.76 2.11 4.11

4.65 2 4 1.16 2.90 5.39

4.25 1 4 1.04 2.50 4.423.88 2 5 1.43 3.20 5.55

3.63 3 6 1.87 3.98 6.78

3.38 2 6 1.72 3.56 5.82

3.21 3 7 2.15 4.27 6.91

3.09 4 8 2.62 5.02 8.10

2.85 4 9 2.90 5.33 8.26

2.60 5 11 3.68 6.40 9.56

2.44 5 12 1.00 6.73 9.77

2.32 5 13 4.35 7.06 10.08

2.22 5 14 4.70 7.52 10.45

2.12 5 16 5.39 8.40 11.41

Approximate values of np

p2/p1 Ac1 Ac2

Grubbs Table for constructing double sampling plans

n1 = n2 , = 0.05, = 0.10

Re1 = Re2 = Ac2 + 1

p1 = AQL, p2 = LTPD


47/47

Home Work

AOQ, and ATI for double sampling plans

acceptance samplng

Documents