Transcript IENG 486 Lecture 14 - Operating Curve Characteristics

IENG 486 - Lecture 14
X-bar & s Charts:
Trial Limits & Standard Limits,
Control Chart Operating Characteristics
7/17/2015
IENG 486: Statistical Quality & Process
Control
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Assignment
 Reading:



CH 8
8.1 – 8.3.2
8.3.4
8.7.1
 Homework:

CH 6 Textbook Problems:


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1; 6 a, b, d only – use Spreadsheet Template (Mat’ls pg)
11; 20 a only; 24; 30
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X-Bar & R-Charts
 The X-Bar Chart
checks variability in
location between
samples
 The R-Chart checks for
changes in sample
variation
UCL
UCL
x
R
LCL
LCL
Sample Number
X-Bar (Means) Control Chart
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Sample Number
R - (Range) Control Chart
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X-Bar & Sigma-Charts
Used
when sample size is greater than 10
 X-Bar Control Limits:

Approximate 3 limits are
found from S & table
 Sigma-Chart Control Limits:

Approximate, asymmetric 3
limits from S & table
UCL  x  A 3 S
UCL  B4 S
CL  x
CL  S
LCL  x  A 3 S
LCL  B3 S
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X-Bar & Sigma-Charts
Limits can
also be generated from historical data:
 X-Bar Control Limits:

 Sigma-Chart Control Limits:
Approximate 3 limits are found
from known 0 & table

Approximate, asymmetric 3
limits from 0 & table
UCL  μ  Aσ
CL  μ
UCL  B 6 
LCL  μ  Aσ
LCL  B 5 
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CL  c 4 
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Operating Characteristic
(OC) Curve
 Ability of the x and R charts to detect shifts (sensitivity) is
described by OC curves
 For x chart; say we know 

Mean shifts from
m0
(in-control value) to
m1 = m0 +k (out-of-control value)
 The probability of NOT detecting the shift on the first sample
after shift is
  P LCL  x  UCL m  m1  m0  k 
 LCL  m1
UCL  m1 
  P
z

 n 
  n
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Ex. Probability of NOT
Detecting Shift
 A 3-sigma x chart is used to monitor a normally distributed
quality characteristic. The process std dev is 1.2 and the
sample size is 5. The process mean is in-control at 22.

Find the probability that a shift to 24.4 is not detected on the first
sample after the shift.
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OC Curve for x Chart
 Plot of  vs. shift size (in std dev units) for various sample
sizes n
OC Curve for x-bar chart with 3-sigma limits
1.00
0.90
0.80
Beta
0.70
0.60
0.50
n=20
n=5
0.40
n=2
n=1
0.30
0.20
0.10
4.
8
4.
4
4.
0
3.
6
3.
2
2.
8
2.
4
2.
0
1.
6
1.
2
0.
8
0.
4
0
0.00
k
 x chart not effective for small shift sizes, i.e., k  1.5
 Performance gets better for larger n and L or larger shifts
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OC curve for R Chart
 Uses distribution of relative range r.v., i.e.,
W R 
 Suppose


0 - in-control std dev
1 - out-of-control std dev
 OC curve for R chart plots  vs. ratio of in-control to out-ofcontrol standard deviation for various sample sizes

That is, plot β vs. l  1/0
 R chart not very effective for detecting shifts for small
sample sizes
(see Fig. 5-14 in text)
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Probability of Detecting Shift
for Subsequent Samples
 After the shift has occurred:






P(NOT detecting shift ON 1st sample)
  0.07078
P(DETECTING shift ON 1st sample)
1     0.93
P(DETECTING shift ON 2nd sample)
 1     0.066
P(DETECTING shift ON rth sample)
 r 1 1   
P(DETECTING shift BY 2nd sample)
1      1     0.93  0.066  0.996
P(DETECTING shift BY rth sample)

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i 1

1   
i 1
r
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Average Run Length (ARL)
 Expected number of samples taken before shift is detected is
called the Average Run Length (ARL)

ARL   r 
r 1
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r 1
1
1    
1   
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Performance of Any
Shewhart Control Chart
 In-Control ARL:

Average number of points plotted on control chart before a false
alarm occurs
(ideally, should be large)
ARL0 
1

 Out-of-Control ARL:

Average number of points, after the process goes out-of-control,
before the control chart detects it
(ideally, should be small)
ARL1 
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1
1 
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ARL Curve for x Chart
3.0
2.9
2.7
2.6
2.4
2.3
2.1
2.0
1.8
1.7
n=1
1.5
1.4
n=2
1.2
1.1
0.8
n=4
0.6
0.5
0.3
0.2
n=20
0.9
20.00
18.00
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
0.0
ARL to detect shift
 Plot of ARL1 vs. shift size (in sd units) for various sample
sizes n:
ARL for x-bar chart with 3-sigma limits
k (shift size)
 Average Time to Signal, (ATS):

Number of time periods that occur until signal is generated on control
chart
ATS   ARL  h 

h - time interval between samples
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Next Assignment
 Reading:




CH 7, CH 6
8.1 – 8.3.2,
8.3.4,
8.7.1
Get Tables 8.2 & 8.3 for your engineering notebook
 Homework:

CH 8 Textbook Problems:

7/17/2015
9, 10, 25
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