10 Sampling Tech 2

8/8/2019 10 Sampling Tech 2

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Introduction

toLot-by-lot

Acceptance SamplingTechniques

by

Attributes


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Topic Outcome:

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

Discuss the properties of an operating-characteristic

(OC) curve.

Design, construct, and use an OC curve.


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Topic Outline:

Operating Characteristics (OC) Curve [LengkokCiri-ciri Pengendalian]

Introduction.

Methods of calculating the probability of acceptance.

Construction of an OC curve using Poisson

Distribution.

Different between Type A and B OC curves.

Properties of an OC curve


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Operating-Characteristic (OC)

Curve An Introduction

An OC curve is a graph ofLot Nonconforming(orPercentNonconforming, 100p0) versus Probability that a sampling

plan would accept the lotsaccep

t the lots, Pa (orPercent of Lots

Accepted, 100Pa).

Material with 0 nonconforming

Accepted always

Pa = 1.0

Material with 100% nonconforming

Rejected always

Pa = 0

100p0

100P

a


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What is the usage of OC curves?

It shows the chancechance ofa lot beingacceptedaccepted for a particularincoming processincoming

process

quality

quality.

It shows the discriminatory power of a samplingplan.


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Ideal OC Curve

1.0

0.5

05.0 10.0

Pro

bab

ilityofAcc

ep

tance,P

a

[orPer

centofLotsA

ccepted,100

Pa

]

Percent Nonconforming (100p0)

Acceptanc

e

Regio

n

Rejec

tion

Regio

n

All lots >5%

nonconforming have

a probability ofacceptance of 0.

All lots


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In actual practice, no sampling plan exists that can be

discriminate perfectlydiscriminate perfectly.

There is always a risk of rejecting a good lot andaccepting a bad lot.

The best we can do is to control the risks.


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Non-Ideal OC Curve

1.0

0.5

05.0 10.0

Pro

bab

ilityofAcc

ep

tance,P

a

[orPer

centofLotsA

ccepted,100

Pa

]



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Summary of common probability distributions

Probability Distributions

Discrete Continuous

Uniform

Binomial

Pascal (negative binomial)

Geometric

Hypergeometric

Uniform

Normal

Exponential

Gamma

Erlang

Weibull

Possion


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Methods of Calculating the Probability of Acceptance:

For attribute sampling, the following distributions are used to

calculate the probability of acceptance.Distribution Formula Conditions

Hypergeometric 1) Population isFINITE.

2) Random sample istaken withoutreplacement.

3) n/N 0.10 canbe approx. by

binomial distribution.

( )

( ) ( )( )

( ) ( )

( )!!

!

!!

!

!!

!

nNn

N

DnDNdn

DN

dDd

D

Nn

DNdn

Dd

dP

C

CC

dP

+

=

=


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Distribution Formula Conditions

Binomial 1) For discreteprobability

distributions that havean infinite number ofitems or that have asteady stream ofitems from a workcenter.

( ) ( )dndqp

dndndP

= 00!!

!


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When these

assumptions are

met, the Poisson

Distribution is

preferable because

of the ease ofcalculation.

Distribution Formula Conditions

Poisson 1) Sample size 16

2) n/N 0.10

3) p0 < 0.1 (on each trial)

( ) ( ) 0!0 np

c

ec

npcP =

Poisson distribution is an excellent approximation to binomial

for almost all sampling plans


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P(d) = probability of d nonconforming units in a sample of size n.

= combinations of all units.

= combinations of nonconforming units.

= combinations of conforming units.

N = number of units in the lot (population).

n = number of units in the sample.

D = number of nonconforming units in the lot.

d = number of nonconforming units in the sample.

N-D = number of conforming units in the lot.

n-d = number of conforming units in the sample.

p0 = proportion nonconforming in the population.

q0 = proportion conforming (1-p0) in the population.

c = count, or number, of events of a given classification

occurring in a sample.

np0 = average count, or average number, of events of a given

classification occurring in a sample.

NnC

DdC

DNdnC


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Construction of an OC curve using PoissonDistribution (Single Sampling Plan)

Lot size, N = 3000Sample size,n = 89

Acceptance number, c = 2

Conditions of using Poisson Distribution:

1) Sample size 16 OK

2) n/N 0.10 0.03

3) p0 < 0.1 (on each trial) ??

Binomial Distribution can be used for simplicityPoisson Distributionis employed.

OBJECTIVE 100p0 vs. 100Pa


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Lot size, N = 3000 Sample size,n = 89 Acceptance number, c = 2

100p0 vs. 100Pa

Assumed ProcessQuality

Probability ofAcceptance

(Poisson

Distribution)

Poisson Tableor

Computer Software

(EXCEL)

p0 = 0.02


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(1) Obtaining Pa value from Poisson Table

np0 = (89)(0.02) = 1.8Acceptance number, c = 2Possible to have 0, 1, or 2 nonconforming units in the sample.

Pa = P0 + P1 + P2

= P2 or less

= 0.731 Pa value is obtained from Poisson

Table for c = 2 and np0 = 1.8

np0 = number of nonconforming


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(2) Obtaining Pa value from EXCEL

Steps:

1) Click icon offx.

2) Function Category: Statistical.

3) Function name: Poisson.

4) Click OK5) x(number of events) = 2

6) Mean (np0) = 1.8

7) Cumulative: Type in TRUE [note: FALSE non-

cumulative]Answer = 0.731

Syntax:

POISSON(x,mean,cumulative)


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Steps of Constructing an OC curve:

1) Assume p0 value

2) Calculate np0 value

3) Attain Pa values from Poisson Table using applicable c

and np0 values or from EXCEL program

4) Plot point (100p0

vs.100Pa)

5) Repeat steps 1 to 4 until a smooth curve is obtained.

Approximately 7 points are needed to describe the curve

with a greater concentration of points where the curvechanges direction.


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p0 100p0 n np0 Pa 100Pa

0 0 89 0 1 1000.0025 0.25 89 0.2225 0.998 99.8

0.005 0.5 89 0.445 0.989 98.9

0.0075 0.75 89 0.6675 0.970 97.0

0.01 1 89 0.89 0.939 93.9

0.0125 1.25 89 1.1125 0.898 89.8

0.015 1.5 89 1.335 0.849 84.9

0.0175 1.75 89 1.5575 0.794 79.4

0.02 2 89 1.78 0.736 73.6

0.0225 2.25 89 2.0025 0.676 67.6

0.025 2.5 89 2.225 0.616 61.6

0.0275 2.75 89 2.4475 0.557 55.7

0.03 3 89 2.67 0.501 50.1

0.0325 3.25 89 2.8925 0.448 44.8

0.035 3.5 89 3.115 0.398 39.8

0.0375 3.75 89 3.3375 0.352 35.2

0.04 4 89 3.56 0.310 31.0

0.0425 4.25 89 3.7825 0.272 27.2

0.045 4.5 89 4.005 0.237 23.7

Assumed Process

QualityProbability of

Acceptance

Number

nonconformin

g


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OC Curve

0

10

20

30

40

50

60

70

80

90

100

0 1.5 3 4.5 6


Per

centofLotsAccepted

(100Pa)

It shows the chance of a

lot being accepted for aparticularincomingincoming

process qualityprocess quality.

e.g.: Incoming process

quality = 2.3%2.3%

66% of the lots is66% of the lots is

expected to beexpected to beaccepted.accepted.


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The above OC curve is unique to the singlesampling plan defined by N = 3000, n = 89, and c = 2.

If this sampling plan does not give the desireddoes not give the desired

effectivenesseffectiveness, then the sampling plan shouldbe changed and a new OC curve should be

constructedconstructed and evaluatedevaluated.


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OC curve for Double Sampling Plans


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OC curve for Multiple Sampling Plans


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Difference between

Type A and Type BOC Curves


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Lot Size

Binomial Distribution

Difference between Type A and Type B OCCurves


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OC Curve Properties


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Sample size as a fixed percentage of lot size


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Fixed sample size

t


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As sample size increases, the curve becomes steeper


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As the acceptance number decreases, the curve

becomes steeper


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END

10 Sampling Tech 2

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