10 sampling tech 2
TRANSCRIPT
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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
]
Percent Nonconforming (100p0)
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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
Percent Nonconforming (100p0)
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