Chapter 7
Attributes Control Charts
許湘伶
Statistical Quality Control
(D. C. Montgomery)
Overview I
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Classify each item inspected: conforming or nonconforming
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attribute variable
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three attributes control charts:
1. p chart; control chart for fraction nonconforming (np chart)
2. c chart; control chart for nonconformities
3. u chart; control chart for nonconformities per unit
Introduction I
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generally not as informative as variables charts
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useful in service industries and nonmanufacturing or
transactional business process
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not easily measured on a numerical scale
C.C. for Fraction Nonconforming I
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fraction nonconforming: the ratio of #{nonconforming
items} in a population to the total number of items in that
population
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The statistical principles underlying the control chart for
fraction nonconforming:
the binomial distribution
I
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p: the probability that any unit will not conform to
specifications
the successive units produced are independent
⇒ each unit Xi ∼ Ber(p)
C.C. for Fraction Nonconforming II
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a random sample of n units of product is selected
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D=
Pn
i=1 Xi :
#{non conforming of units}
⇒ D ∼ B(n, p)
!
P{D = x} =
n x
p (1 − p)n−x ,
x
E(D) = np;
Var(D) = np(1 − p)
x = 0, 1, . . . , n
The sample fraction nonconforming= the ratio of the number of nonconforming units in the sample D to the sample
size n
D
p(1 − p)
⇒ p̂ =
⇒ E(p̂) = µp̂ = p; Var(p̂) = σp̂2 =
n
n
C.C. for Fraction Nonconforming III
Review:
the general model for the Shewhart control chart
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w: statistic that measures a quality characteristic
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µw , σw2 : the mean and variance of w
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L: the distance of the control limits from the center
line (Customary choosing L = 3)
UCL = µw + Lσw
Center line = µw
UCL = µw − Lσw
C.C. for Fraction Nonconforming IV
I
p: the true fraction nonconforming in the production
process (a specified standard value)
Fraction Nonconforming C.C: Standard Given
s
p(1 − p)
n
s
p(1 − p)
n
UCL = µp + Lσp = p + 3
Center line = µp = p
UCL = µp − Lσp = p − 3
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LCL < 0 ⇒ customarily set LCL = 0
⇒ assume the control chart only has an upper control
limits
C.C. for Fraction Nonconforming V
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p is not know ⇒ estimated from observed data
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m preliminary samples, each of size n (m : 20 ∼ 25)
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Di : nonconforming units in sample i
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the fraction nonconforming in the ith sample:
pi =
I
Di
,
n
i = 1, . . . , m
the average of these individual sample fractions
nonconforming:
Pm
p̄ =
(estimates p)
i=1 Di
mn
Pm
=
i=1 p̂i
m
C.C. for Fraction Nonconforming VI
Fraction Nonconforming C.C: No Standard Given
s
UCL = p̄ + 3
p̄(1 − p̄)
n
Center line = p̄
s
UCL = p̄ − 3
p̄(1 − p̄)
n
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the trial control limits based on m initial samples
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p̂i : to test whether the process was in control when the
preliminary data were collected
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Phase I: Any points that exceed the trial control limits
should be investigated.
C.C. for Fraction Nonconforming VII
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If assignable causes for these points are discovered, they
should be discarded and new trial control limits
determined.
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If p is given ⇒ the calculation of trial control limits is
unnecessary
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∵ p would rarely be known, we would be given a standard
value of p (represents a target value for the process)
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If future samples indicated an out-of-control condition, we
must determine whether the process is out of control at the
target p but in control at some other value of p.
Example 7.1: cardboard cans I
Frozen orange juice
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packed in 6 oz cardboard cans
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determine whether it could possibly leak either on
the side seam or around the bottom joint
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Set up a control chart to improve the fraction of
nonconforming cans produced by this machine
Example 7.1: cardboard cans II
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m = 30 samples of n = 50 cans
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selected at 1/2 hour intervals over a three-shift(三班制)
period
Example 7.1: cardboard cans III
Example 7.1: cardboard cans IV
Example 7.1: cardboard cans V
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Sample 15 & 23:
investigated whether
an assignable
cause??
Example 7.1: cardboard cans VI
Example 7.1: cardboard cans VII
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Excluding assignable
causes (Samples 15
& 23)
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Analysis of the data
does not produce
any reasonable
assignable cause for
sample 21
⇒ decide to retain
the point
Example 7.1: cardboard cans VIII
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If the new operator
working during the entire 2
hors period (samples 21-24)
⇒ we should discard all
four sample (21-24)
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p̄ is much too high⇒
Engineering staff: several
adjustments can be made
on the machine
Example 7.1: cardboard cans IX
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no assignable cause of this
out-of-control signal can be
determined
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Test: the process fraction
nonconforming in this
current three-shift period
differs from the previous
one
H0 : p1 = p2
Example 7.1: cardboard cans X
The hypotheses:
H0 : p1 = p2
H1 : p1 > p2
P54
p̂1 = p̄ = 0.2150;
p̂2 =
i=31 Di
(50)(24)
= 0.1108
The (approximate) test statistic (p.145) :
p̂1 − p̂2
Z0 = q
= 7.10 > Z0.05 = 1.645
p̂(1 − p̂)( n11 + n12 )
p̂ =
n1 p̂1 + n2 p̂2
= 0.1669
n1 + n2
⇒ reject H0 i.e., there has been a significant decrease in the
process fallout
Example 7.1: cardboard cans XI
The new control chart:
UCL = 0.2440
Center line = p̄ = 0.1108
LCL = max{−0.0224, 0} = 0
Example 7.1: cardboard cans XII
Fraction Nonconforming Control Chart I
Three parameters in the fraction nonconforming control chart
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the sample size (n)
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the frequency of sampling
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the width of the control limits
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Common to base a control chart for fraction nonconforming
on 100% inspection of all process output over some
convenient period of time
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I
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a shift (一個輪班)
a day
interrelated between sample size and sampling frequency
Fraction Nonconforming Control Chart II
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sampling frequency
1. appropriate sampling frequency for the production rate (⇒
fixes n)
2. rational subgrouping
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Ex: three shifts-suspect shifts differ in their general quality
level
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each shift as a subgroup;
Fraction Nonconforming Control Chart III
Sample size:
I
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p is very small
⇒ choose n sufficiently large
⇒ a high probability of finding at least one nonconforming
unit in the sample
Otherwise, find the control limits: the presence of only one
nonconforming unit in the sample would indicate an
out-of-control condition
Ex: (p, n) = (0.01, 8) ⇒ UCL = p + 3
one nonconforming unit in the sample
⇒ p̂ = 81 = 0.1250
⇒ out of control
q
p(1−p)
n
= 0.1155
Fraction Nonconforming Control Chart IV
作法(1)：
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choose n s.t.
P(finding at least on nonconforming unit per sample) ≥ γ
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D: #{nonconforming items in the sample}
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Ex: p = 0.01 ⇒ P(D ≥ 1) ≥ 0.95
1. The binomial distribution:
⇔P(D = 0) =
n!
(0.01)0 (1 − 0.01)n−0 = 0.05
0!(n − 0)!
⇒n = 298
2. Using the Poisson approximation to the binomial
distribution
λ = np ≥ 3 ⇒ n = 3/p = 300
Fraction Nonconforming Control Chart V
作法(2)：
I
I
Duncan (1986): choose n large enough s.t. approximately a
50% chance of detecting a process shift of some specified
amount
p0 = 0.01 ⇒ a shift to p1 = 0.05
P(detect the shigt) ≥ 0.05
r
⇔P
|p̂ − p0 | > L

⇔P 
p0 − p1 − L
p
q
!
p0 (1 − p0 ) p1
n
p0 (1−p0 )
n
p1 (1 − p1 )/n
r
⇔p1 ≈ p0 + L
≥ 0.5
p0 − p1 + L
p̂ − p1
< q
<
q
p1 (1−p1 )
n
p0 (1 − p0 )
(Regard
n
p0 −p1 −L
√
q
p0 (1−p0 )
n
p1 (1−p1 )
n
√
p0 (1−p0 )/n
p1 (1−p1 )/n
≈ 0)

 = 0.5
Fraction Nonconforming Control Chart VI
s
δ = p1 − p0 ⇔ δ = L
⇒n=
2
L
δ
p0 (1 − p0 )
n
p0 (1 − p0 )
⇒(p0 , δ, L) = (0.01, 0.04, 3) ⇒ n = 56
作法(3)：
I
In control and p is small ⇒ choose n s.t. LCL > 0
s
LCL = p − L
p(1 − p)
(1 − p) 2
>0⇔n>
L
n
p
(p, L) = (0.05, L) ⇒ n > 171
Fraction Nonconforming Control Chart VII
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Three-sigma control limits are usually employed on the p
chart
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the fraction nonconforming control chart is not a
universal(通用的) model for all data on fraction
nonconforming
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based on binomial distribution:
(i) p = constant
(ii) successive(連續的) unit of production are independent
nonconforming units: clustered or dependent
⇒ the fraction nonconforming control chart is often of
little use
Fraction Nonconforming Control Chart VIII
Care:
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interpreting points that plot below the LCL
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not represent a real improvement in process quality (ex:
caused by errors; improperly calibrated test and inspection
equipment)
The np Control Chart I
(不良品個數)
number nonconforming (np) control chart
q
UCL = np + 3 np(1 − p)
Center Line = np
q
LCL = np − 3 np(1 − p)
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p: unavailable ⇒ p̂ = p̄
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np chart: easier to interpret than p chart
The np Control Chart II
Example 7.2: an np control chart
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the orange juice concentrate can process in Table 7.1
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p̄ = 0.2313; n = 50
UCL = 20.510
Center Line = 11.565
LCL = 2.620
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#{nonconforming units}: integer
⇒ (LCL,UCL)=(2,21)
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if a sample value of np plotted at or beyond the
control limits
The np Control Chart III
Variable Sample Size I
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the sample is a 100% inspection of process output over
some period of units
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different numbers of units in each period
Three approaches to constructing and operating a control chart
with a variable n
1. Variable-width control limits
2. Control limits based on an average sample size
3. The standardized control chart
Variable Sample Size II
Variable-width control limits:
P25
Di
i=1 ni
p̄ = Pi=1
25
s
UCL = p̄ + 3σ̂p̂ = p̄ + 3
s
LCL = p̄ − 3σ̂p̂ = p̄ + 3
p̄(1 − p̄)
ni
p̄(1 − p̄)
ni
Variable Sample Size III
Variable Sample Size IV
Control limits based on an average sample size:
P25
n̄ =
i=1 ni
25
s
UCL = p̄ + 3σ̂p̂ = p̄ + 3
s
LCL = p̄ − 3σ̂p̂ = p̄ + 3
p̄(1 − p̄)
n̄
p̄(1 − p̄)
n̄
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Assume: future sample sizes will not differ greatly
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unusually large variation in the size of a sample or a point
plots near the approximate control limits
⇒ the exact control limits for that point should be
determined
Variable Sample Size V
Variable Sample Size VI
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Sample 11: close to the
UCL, yet in control
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But is using
Variable-width control
limits: it is
out-of-control
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Care: points near the
approximate control
limits
Variable Sample Size VII
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Problem: a change in p̂i must be interpreted relative to ni
p = 0.20 and two successive sample
p̂i = 0.28, ni = 50
p̂i+1 = 0.24, ni+1 = 250
⇒p̂i > p̂i+1 ?
0.28 − 0.2
0.24 − 0.2
∵p
= 1.41 < 1.58 = p
0.2(1 − 0.2)/50
0.2(1 − 0.2)/50
Variable Sample Size VIII
The standardized control chart:
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the points are plotted in standard deviation units
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Center line = 0; (UCL, LCL)=(+3,-3)
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The variable in standardized control chart:
p̂i − p
Zi = q
p(1−p)
ni
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p: the process fraction nonconforming in the in-control state
if no standard is given p̂ = p̄
Variable Sample Size IX
Minitab: Download STANDARD.mac
Variable Sample Size X
%STANDARD ’Di’ ’ni’
Variable Sample Size XI
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difficult for operating personnel to understand and
interpret
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the actual process fraction defective has been “lost”
OC function and ARL I
Operating-characteristic (OC) function:
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a graphical display of the probability of incorrectly
accepting the hypothesis of statistical control (type II
error, β-error) vs. p
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providing a measure of the sensitivity of the control
ability to detect a shift from the nominal value p̄ to
some other value p
β = P{p̂ < UCL|p} − P{p̂ ≤ LCL|p}
= P{D < nUCL|p} − P{D ≤ nLCL|p} (D ∼ B(n, p))
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if LCL<0 ⇒ P{D ≤ nLCL|p} should be dropped
OC function and ARL II
OC function and ARL III
ARL(平均連串長度)
ARL =
1
P{sample point plots out of control}
1
α
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In control: ARL0 =
I
Out of control: ARL1 =
1
1−β
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p = p̄: α = 1 − β = 0.0027
1
⇒ ARL0 = 0.0027
≈ 370
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p = 0.3: β = 0.8594
1
⇒ ARL1 = 1−0.8594
=7
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n ↑⇒ β ↓⇒ ARL1 ↓
Control charts for defects I
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A nonconforming item: a unit of product
I
I
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does not satisfy one or more of the specifications of the
product
at least one nonconformity
depending on their nature and severity(嚴重): possible
contain several nonconformities and not be classified as
nonconforming
1. c chart: total number of nonconformities in a unit
2. u chart: average number of nonconformities per unit
Assume: the occurrence of nonconformities in samples of
constant size ∼ Poisson distribution
c chart I
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defects∼ Poisson distribution
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x: #{nonconformities}
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c: the parameter of the Poisson distribution
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three-sigma limits
x ∼ P(c) ⇒ p(x) =
e −c c x
,
x!
x = 0, 1, 2, . . .
c chart: Standard Given
√
UCL = c + 3 c
Center line = c
√
LCL = c − 3 c
(if obtaining nagtive LCL⇒LCL=0)
c chart II
c chart: No Standard Given (trial control limits)
√
UCL = c̄ + 3 c̄
Center line = c̄
√
LCL = c̄ − 3 c̄
c̄ = the observed average number of nonconformities in a
preliminary sample of inspection units
c chart III
Example 7.3: Printed Circuit Boards
I
26 successive sample of 100 boards
c̄ = 19.85,
√
√
(UCL, LCL) = (c̄+3 c̄, c̄−3 c̄) = (33.22, 6.48)
c chart IV
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two points plot outside the control limits
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exclude the two samples:
c̄ = 19.67,
(UCL, LCL) = (32.97, 6.36)
c chart V
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#{nonconformities per board} is still unacceptably high
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further action is necessary to improve the process
c chart VI
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c chart is more informative than p chart
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several different types of nonconformities
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analyzing the nonconformities by type⇒ insight into their
cause
c chart VII
u chart I
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the control chart on a sample size of n inspection units
(1) nc chart:
√
UCL = nc̄ + 3 nc̄
Center line = nc̄
√
LCL = nc̄ − 3 nc̄
(2) u chart: the average number of nonconformities per
inspection unit i.e., u = nx
r
UCL = ū + 3
ū
n
Center line = ū
r
LCL = ū − 3
ū
n
u chart II
Variable sample size: the number of inspection units in a
sample will not be constant
u chart III
1. Based on an average sample size: n̄ =
2. standardized statistic:
ui −ū
Zi = q
⇒ (UCL, LCL) = (+3, −3)
ū
ni
Pm
i=1 ni /m
Alternative probability models for count
data I
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c chart: assume the Poisson distribution
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nonconformities: cluster patterns
I
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compound Poisson distribution
Mixtures of various types of nonconformities
Demerit Systems I
demerit(缺點) scheme:
The number of demerits in the inspection unit:
di = 100ciA + 50ciB + 10ciC + ciD
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demerit weights for Class (A,B,C,D): (100,50,10,1)
Demerit Systems II
I
#{demeritPper unit}:
n
Di
i=1
ui = D
∼
n =
n
linear combination of independent Poisson r.v.
UCL = ū + 3σ̂u
Center line = ū
LCL = ū − 3σ̂u
whereū = 100ūA + 50ūB + 10ūC + ūD
"
(100)2 ūA + (50)2 ūB + (10)2 ūC + ūD
σ̂u =
n
#1/2
Demerit Systems III
OC curve of u chart:
β = P{x < UCL|u} − P{x ≤ LCL|u}
= P{c < nUCL|u} − P{cx ≤ nLCL|u}
= P{nLCL ≤ x < nUCL|u}
dn UCLe
=
X
x=<n LCL>
e −nu (nu)x
x!
Attributes and Variables Control Chart I
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Variables control chart: x̄ and R charts, x̄ and s charts
I
I
I
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more information about process performance
process mean and variability might be obtained
provide relative to the potential cause of that out-of-control
signal
indication of impending(即將發生的) trouble and allow to
take corrective action before any defectives are actually
produced
Attributes and Variables Control Chart II
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Attributes control chart: p (np) chart, c chart, u chart
I
I
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several quality characteristics can be considered jointly
sometimes avoiding expensive(昂貴的) and
time-consuming(費時的) measurements
Generally, variables C.C. are preferable to attributes
Attributes and Variables Control Chart III
Example 7.7: Advantage of Variables C.C.
I Nominal value of the mean and std: (µ, σ) = (50, 2)
I SL (±3-σ): (USL,LSL)=(56,44)
I x̄ chart: the process is in control at the nominal level of µ0 50,
p0 = 0.0027
I Suppose: µ0 = 50 → µ1 = 52; p0 = 0.0027 → p1 = 0.0228;
β = 0.50
i.e., P{detecting this shift on the first subsequent sample} = 0.50
I Appropriate n for x̄ chart and comparing it to the n for a p
chart?
Attributes and Variables Control Chart IV
3(2)
UCL of x̄ chart: 50 + √ = 52 ⇒ n = 9
n
β-error of p chart: n =
2
L
δ
p0 (1 − p0 ) = 59.98∼
= 60
Attributes and Variables Control Chart V
Example 7.8: Misaaplication of x̄ and R charts
I
inspected a sample of the production units several
times each shift using attributes inspection
I
p̂i : estimate of the process fraction nonconforming
I
A consultant: converting their fraction
nonconforming data into x̄ and R charts;
each group of 5 successive values of p̂i
5
1X
x̄ =
p̂i ;
5 i=1
R = max(p̂i ) − min(p̂i )
Attributes and Variables Control Chart VI
Attributes and Variables Control Chart VII
Guidelines for Implementing C.C. I
Some general guidelines helpful in implementing control chart:
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Determining which process characteristics to control
I
Determining where the charts should be implemented in
the process
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Choosing the proper type of control charts
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Taking actions to improve processes as the results of
SPC/control chart analysis
I
Selecting data-collection systems and computer software