Transcript Control Charts for Individuals

IENG 486 - Lecture 17
Control Charts for Individuals
(Measuring Each Unit)
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Assignment
 Reading:

Chapter 6


Section 6.4: pp. 259 - 265
Chapter 9



Sections 9.1 – 9.1.5: pp. 399 - 410
Sections 9.2 – 9.2.4: pp. 419 - 425
Sections 9.3: pp. 428 - 430
 Homework:

CH 9 Textbook Problems:

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1a, 17, 26
Hint: Use Excel charts!
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Individual Measurements
 Sometimes repeated measures in a subsample don't make
sense:


inventory level
accounts payable – price of an item
 Other reasons for using individual measurements



Variation in sample only reflects measurement error, e.g., batch
production of chemicals
Automated inspection – every unit is analyzed
Production rate very slow – inconvenient to wait for large enough
sample
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Control Charts for Individual
Measurements
 Notes about Individuals Charts:



Sample points must be relatively frequent
There is more sampling error (false alarm & insensitivity)
Sample points tend to be non-normal

points are not averages and central limit theorem does not apply
 Control Chart Types for Individuals:




Shewhart x-chart and Moving Range chart
MA – Moving Average chart
EWMA – Exponentially Weighted Moving Average
CUSUM – Cumulative Sum
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Moving Range Control Chart
 MR2i = |xi – xi-1|
MR3i = |xi – xi-2|
or
 Computation of the Moving Range:
Obs i
xi
MR2
MR3
1
33.75
-
-
2
33.05
0.70
-
3
34.00
0.95
0.25
4
33.81
0.19
0.76
5
33.46
0.35
0.54
…
…
…
…
n
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Moving Range Control
Chart, Cont'd
 General model for moving range chart:
UCL  m MR  Ls MR
CL  m MR
LCL  m MR  Ls MR
 Plot MR2i = |xi – xi-1| or MR3i = |xi – xi-2| on control chart
 Substituting estimates for mR and sR and using “3-sigma”
limits:
CL  m MR  MR
 Where MR is:
MR 2 
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n
1
n1
 MR2
i 2
i
or
MR3 
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n
1
n2
 MR3
i 3
i
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Moving Range Control
Chart, Cont'd
 For MR2 use d2 for n = 2
For MR3 use d2 for n = 3
UCL  m MR  3s MR  MR  3  d3s 
 MR  3d3
MR
 D4 MR
d2
LCL  D3 MR  0
 Very similar to Range chart except we’re using moving range
instead of average range
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x Control Chart
(Individual Measurements Chart)
 Plot sample statistic: x
 General model for x chart
UCL  m x  Ls x
CL  m x
LCL  m x  Ls x
 Substituting estimates for mx and sx and using 3-sigma limits
CL  m x  x
where:
mx  x 
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n
1
n
x
i 1
i
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x Control Chart
Cont'd
 x chart upper and lower limits:
UCL  m x  3s x  x  3s
MR
 x 3
d2
MR
LCL  x  3
d2
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Cautions for x & Moving
Range charts:
 Always check x’s for normality

If x’s not normal, control limits are inappropriate
 Use zone rules ONLY if the x’s are Normal
 Very BAD at detecting small shifts, i.e., shifts < 2s
x chart
(n = 5)
b
size of shift
x chart
(n = 1)
b
ARL1
ARL1
1s
0.78
5.25
0.98
43.96
2s
0.07
1.08
0.84
6.30
3s
0.00
1.00
0.50
2.00
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Montgomery(5th ed.) Example 5-5, p. 250
Viscosity of Aircraft Primer Paint
 Plot MR2i = |xi – xi-1| on control chart
 From initial data compute x and MR:
mˆ  x  33.52
MR  0.48
sˆ 
MR
0.48

d2
1.128
(since d2 = 1.128 for n =2)
 Control Limits for Moving Range chart (use D4 & D3 for n =2)
UCL  D4 MR  3.267  0.48  1.57
CL  MR  0.48
LCL  D3 MR  0
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Ex.: Viscosity of Aircraft Primer Paint, Cont'd
UCL  x  3sˆ  34.80
CL  x  33.52
LCL  x  3sˆ  32.24
 Compute control Limits for x Chart
X Chart for Viscosity
X
35
UCL = 34.80
34.5
CTR = 33.52
34
LCL = 32.24
33.5
33
32.5
32
0
3
6
9
12
15
Observat ion
MR(2) Chart for Viscosity
1.6
UCL = 1.57
CTR = 0.48
MR(2)
1.2
LCL = 0.00
0.8
0.4
0
0
3
6
9
12
15
Observat ion
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Alternatives to Shewhart
Control Charts
 All control charts so far have been Shewhart Control Charts


uses information about the process contained in the last plotted point
ignores information given by the entire sequence of points, unless
sensitizing rules are used
 Shewhart charts are relatively insensitive to small shifts,
ex. shifts < 1.5s
 Three Alternative charts:



MA – Moving Average control chart
EWMA – Exponentially Weighted Moving Average control chart
CUSUM – Cumulative-sum control chart
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MA Control Chart
(Non-Shewhart Control Chart)
 Plot sample statistic: average of last w data points (Mi )
 Computing point to plot ( Mi ) for the chart:
Mi 
xi  xi 1  ... xi  w1
w
 Estimate for μ (to find center line):
1 n
μ0   xi
n i 1
 Estimate for s (to find control limits, changes with each point):
σx
σ
w
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MA Control Chart
(Non-Shewhart Control Chart)
 General model for MA control chart
σ
UCL  μ0  Lσ  μ0  L x
w
1 n
CL  μ0   xi
n i 1
σ
UCL  μ0  Lσ  μ0  L x
w
 Notes:



Picking w larger makes chart faster to detect to smaller shifts
Picking w smaller makes chart more sensitive to larger shifts
MA is better at detecting smaller shifts than a Shewhart chart,
but not as effective as a EWMA or CUSUM chart
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EWMA - Exponentially Weighted
Moving Average Control Chart
 The EWMA control chart is good for detecting small shifts
 EWMA can be used to monitor

process mean or variance
 Plot sample statistic:


zi = l (current x)
+ (1 - l)(weighted avg of past x's)
That is: zi = lxi + (1- l)zi -1
ˆ
 Use estimate for z0  m
OR target value z0  m 0
 l is weighting factor, where 0 < l < 1
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Example Computing
EWMA Statistic
 Process mean is 14.31. Here are the first three observations.
 Compute the EWMA statistic, zi, with weight l = 0.2.
Obs i
xi
1
14.56
z1 = (0.2)(14.56) +(0.8)(14.31)
= 14.36
2
13.88
z2 = (0.2)(13.88) +(0.8)(14.36)
= 14.26
3
13.98
z3 = (0.2)(13.98) +(0.8)(14.26)
= 14.20
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EWMA
zi = lxi + (1- l)zi-1
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EWMA Control Limits
 Standards Given: UCL  m z  Ls z
CL  m z
LCL  m z  Ls z
 Standards not given - use estimates for mz and sz:
mˆ z  x
sˆ z  sˆ
l
1  1  l 2i 

2  l  
 Notice: σ depends on the observation number i

Use sˆ z to estimate s: σ
ˆz
 Typical values for l and L:

0.05  l  0.25
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and
2.6  L  3.054
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Example 9.2, p. 435:
EWMA Chart for Process Mean
 Set up EWMA chart for following data from a process with
mean 10 and std dev 1. Use l = 0.1 and L = 2.7.
Obs i
xi
1
9.45
2
7.99
3
9.29
…
…
30
10.52
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EWMA
zi = lxi + (1- l)zi-1
Z1=9.945 UCL=10.27 LCL=9.73
Z2=9.799 UCL=10.36 LCL=9.63
Z3=9.929 UCL=10.42 LCL=9.58
…
Z30=10.052 UCL=10.62 LCL=9.38
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Example:
EWMA Chart Cont'd
EWMA Chart for x
10.8
UCL = 10.62
EWMA
10.5
CTR = 10.00
LCL = 9.38
10.2
9.9
9.6
9.3
0
5
10
15
20
25
30
Observation
 Two points above  Process is out-of-control
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Design of EWMA Control Chart:
How to pick l and L
 Use smaller l to detect smaller shifts
 Usual choices:

l  0.05, l  0.10, l  0.20
 Reasonable configuration:

For l  0.20 let L = 3
 For smaller l, use slightly smaller L


For l  0.05 let L  2.6
For l  0.10 let L  2.8
 See Table 9.11, p.437 for ARL’s
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CUSUM Control Chart
 Incorporates all the information in the sequence of sample
values by

plotting the cumulative sums of the deviations of the sample values
from a target value, m0
 CUSUM can be used to monitor




process mean
defectives
defects
variance
 CUSUM can have sample size n  1
 We concentrate on sample size n = 1
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Basic Principle of CUSUM
 Plot Ci – CUSUM sample statistic
Ci    x j  m 0 
i
j 1
 Example: Say target m0 = 10
Obs i
xi
(xi – m0)
Ci    x j  m0 
i
j 1
1
9.45
-0.55
-0.55
2
7.99
-2.01
-0.55 – 2.01 = -2.56
3
9.29
-0.71
-2.56 – 0.71 = -3.27
4
11.66
1.66
-3.27 + 1.66 = -1.61
5
12.16
2.16
-1.61 + 2.16 = 0.55
 If the process remains in-control, Ci remains near 0
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Tabular CUSUM Control Chart
 xi ~ N(m0, s) - quality characteristic
 CUSUM works by compiling the statistics:


Ci+ = accumulated deviations above m0 (resets to 0 if it would go negative)
Ci– = accumulated deviations below m0 (resets to 0 if it would go negative)
 The Tabular CUSUM

Record following values in table:
Ci   max 0, xi   m 0  K   Ci 1 
Ci   max 0,  m 0  K   xi  Ci 1 

where starting values are
C0  C0  0
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Tabular CUSUM Control
Chart Cont'd
 Let m1 = out-of-control value then
m1  m 0
K 

2
K is reference value chosen halfway between target m0 and out-ofcontrol value
 With shift expressed in std dev units, i.e.,
m1  m0  s  s  m1  m0
s
K


2
Ci  and Ci  accumulate deviations from μ0 that are greater than K
Ci  and Ci  are reset to zero upon becoming negative
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How to Determine if Process
Out-of-Control?
 H - decision interval


C
C
i
i
 If or exceed the decision interval (H), the process is
considered out-of-control
 Rule of thumb value for H

Choose H to be five times the process standard deviation, H = 5s
 Counters N+ and N– record the number of consecutive
periods the CUSUM Ci and Ci rose above zero, respectively.

The counters can be used to indicate when the shift most likely
occurred
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Example 9.1, p. 418
 m0 =10, n =1, s = 1.0
 Say magnitude of shift we want to detect is s = 1 (1.0) = 1.0
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Period i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
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xi
9.45
7.99
9.29
11.66
12.16
10.18
8.04
11.46
9.20
10.34
9.03
11.47
10.51
9.40
10.08
9.37
10.62
10.31
8.52
10.84
10.90
9.33
12.29
11.50
10.60
11.08
10.38
11.62
11.31
10.52
 xi
 10.5
-1.05
-2.51
-1.21
1.16
1.66
-0.32
-2.46
0.96
-1.30
-0.16
-1.47
0.97
0.01
-1.10
-0.42
-1.13
0.12
-0.19
-1.98
0.34
0.40
-1.17
1.79
1.00
0.10
0.58
-0.12
1.12
0.81
0.02
Ci 
0.00
0.00
0.00
1.16
2.82
2.50
0.04
1.00
0.00
0.00
0.00
0.97
0.98
0.00
0.00
0.00
0.12
0.00
0.00
0.34
0.74
0.00
1.79
2.79
2.89
3.47
3.35
4.47
5.28
5.30
N
 9.5  xi 

0
0
0
1
2
3
4
5
0
0
0
1
2
0
0
0
1
0
0
1
2
0
1
2
3
4
5
6
7
8
0.05
1.51
0.21
-2.16
-2.66
-0.68
1.46
-1.96
0.30
-0.84
0.47
-1.97
-1.01
0.10
-0.58
0.13
-1.12
-0.81
0.98
-1.34
-1.40
0.17
-2.79
-2.00
-1.10
-1.58
-0.88
-2.12
-1.81
-1.02
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Ci 
0.05
1.56
1.77
0.00
0.00
0.00
1.46
0.00
0.30
0.00
0.47
0.00
0.00
0.10
0.00
0.13
0.00
0.00
0.98
0.00
0.00
0.17
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
N

1
2
3
0
0
0
1
0
1
0
1
0
0
1
0
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
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Tabular CUSUM Example
Control Chart for ________________________
Ci+
Ci-
Xi-Mu
H+
H-
6.000000
4.000000
CUSUM
2.000000
0.000000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
-2.000000
-4.000000
-6.000000
Scale
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Sample Number
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Estimate of New Shifted
Process Mean

Ci 
m0  K   ,


N
mˆ  

C
m  K  i ,
0

N

if Ci   H
if Ci   H
 Use this estimate to bring process back to the target value m0

 e.g.: At period 29, C29  5.28
 New process average estimate is
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Notes about CUSUM control
charts
 Do not apply zone rules
 Do not apply run rules


 Successive values of Ci and Ci are not independent
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Recommendations for
CUSUM design
 Let H = hs and K = ks where s is process std dev
 Using h = 4 or h =5 and k = 1/2 gives CUSUM w/ good ARL
Shift (multiple of s)
ARL for h = 4
ARL for h = 5
0
168
465
0.25
74.2
139
0.50
26.6
38.0
0.75
13.3
17.0
1.00
8.38
10.4
1.50
4.75
5.75
2.00
3.34
4.01
2.50
2.62
3.11
3.00
2.19
2.57
4.00
1.71
2.01
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Guidelines for Implementing
Control Charts
1.
Determine which process or product characteristic(s) to
control
2.
Determine where the charts should be implemented in
process
3.
Choose proper type of control charts
4.
Decide what actions should be taken to improve processes
5.
Select data-collection systems and computer software
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Determine Which Characteristic to
Control and Where to Put Charts
1.
2.
3.
4.
To start, apply charts to any process or product
characteristics believed important.
Charts found unnecessary are removed; others that may be
required are added.
(Usually more charts to start than after process has
stabilized)
Keep current records of all charts in use, i.e., types and
parameters of each.
If charts used effectively  number of charts for variables
increases and number of attributes charts decreases
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Determine Which Characteristic to
Control and Where to Put Charts
5.
6.
At beginning, use more attributes charts applied to finished
units, i.e., near end of process.
As more is learned about the process, these are replaced
with variables charts earlier in process on critical process
characteristics that affect nonconformities.
Rule of thumb: the earlier in the process that control can be
established, the better.
Control charts are an on-line process monitoring
procedure; Maintain charts as close to work center as
possible.
Operators and process engineers should be directly
responsible for using, maintaining and interpreting charts
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Choosing Proper Type of
Control Chart: Variables Charts

Use (x & R) or (x & S) charts when:
1.
2.
3.
4.
5.
6.
7.
8.
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New process or product coming online
Chronically troubled process
Wish to reduce downstream acceptance sampling
Using attributes charts but yield still unacceptable
Very tight specifications
Operator decides whether or not to adjust process
Change in product specs desired
Process capability (stability) must be continually
demonstrated
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Choosing Proper Type of
Control Chart: Attributes Charts

Use p, np, c or u charts when:
1.
2.
3.
4.
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Operators control assignable causes and it is necessary to
reduce fallout
Process is complex assembly operation and product quality
measured in terms of occurrence of nonconformities: e.g.
computers, automobiles
Measurement data cannot be obtained
Historical summary of process performance is necessary.
Attributes charts are effective for summarizing a process for
management
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Choosing Proper Type of
Control Chart: Individuals Charts

Use (x & MR), MA, EWMA, or CUSUM charts when:
1.
2.
3.
4.
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Repeated measures do not make sense
Inconvenient / impossible to obtain more than one
measurement per sample
Automated testing allows you to measure every unit
(EWMA chart may be best)
Data becomes available very slowly and waiting for a larger
sample is impractical.
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