Download Multi-stream Pocess Control

A Process Control Screen for
Multiple Stream Processes
An Operator Friendly Approach
Richard E. Clark
Process & Product Analysis
Multiple Stream Processes
• Injection Molding
• Extrusion Blow
Molding
• Reheat Stretch Blow
Molding
• Thermoforming
• Multilayer Sheet
Extrusion
• Double Seaming
• Filling Machines
• Heat Sealing
Machines
• Labelers
History
Year
1978
1993
Number of Stations
4
48
Production Rate
2400
48000
Number of Characteristics
Monitored
6
10+
Number of Charts Monitored
24
480+
Method of Collection and Analysis
Manual Computer
The Object of This Paper is to
Describe a System of Charts
to Be Used by Operators
and/or Inspectors to Control
Multiple Stream Processes.
The Operator Needs to Know
– That the process is adjusted so that the
average of the characteristic being monitored
is equal to the targeted mean.
– That the means and variation of the individual
streams are being maintained within an
acceptable range.
– That the pattern of variation among streams is
stable.
– That the individual items from all stations are
conforming
to
internal
or
customer
specification limits.
Process Model
Yijk =  + Ti + Pj + k(ij)
i = 1, 2, …, t
j = 1, 2, …, p
K = 1, 2, …, n
 represents the process mean.
Ti is an independently and normally distributed
random variable with mean 0 and variance t2 which
represents the process variation with time. By
definition, TI equals 0 for an in control process.
Pj is a fixed value representing the effect of station j.
In order for the process average to = , the sum of
the Pj over the j stations must be 0.
Process Model (cont.)
k(ij) is an independently and normally distributed
random variable with mean 0 and variance 2
resulting from random variation in the process and
measurement system.
For this paper, 2 is
assumed to be constant for all positions and times.
Observations from an “In Control” 5
Station Machine are Shown in the Table
Below
Station
Value
1
Yi11 =  + 0 + P1 + 1(I,1)
2
Yi21 =  + 0 + P2 + 1(I,2)
3
Yi31 =  + 0 + P3 + 1(I,3)
4
Yi41 =  + 0 + P4 + 1(I,4)
5
Yi51 =  + 0 + P5 + 1(I,5)
Average Computation
The average value for time i is calculated using the
following equation.
_
Yi.. = (5* + P1 + P2 + P3 + P4 + P5 + 1(I,1) + 1(I,2) +
1(I,3) + 1(I,4) + 1(I,5))/5
By definition P1 + P2 + P3 + P4 + P5 = 0 and the expected
values for 1(i,j)’s is 0. Therefore;
_
Yi.. = 
And is an unbiased estimate of the population mean.
Confidence Intervals
The random component in each observation, k(ij), is
independent of other observations and randomly
distributed with mean 0 and variance 2 .
_
Therefore, the confidence intervals for the means and
observations from this process at time i are as follows.
The mean at time i
_
Yi.. ± 3*/√5
The mean for each position is:
_
Y.j. =  + Pj
Confidence Intervals (cont.)
And the confidence intervals for control limits for the
measurements from each position for an “in control”
process are:
_
YK(ij) = Y.j. ± 3*
Distributions Used to Generate Data for Examples
Station
Average
Standard
Deviation
19
19.0
1.00
20
20.0
1.00
21
21.0
1.00
22
22.0
1.00
23
23.0
1.00
Data
Station
Set
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
Average
R-bar MR(2)
19
19.4
18.6
20.6
17.4
21.8
20.6
19
19.4
19.4
18.2
18.6
18.6
19.4
20.2
17
19.8
19.8
18.6
18.2
18.6
18.6
19
17.8
17.8
19
19.4
19.4
19
18.6
19
17.8
20.2
21
18.2
19
19.4
18.2
18.2
17.4
20.6
20.2
18.2
19.4
19.8
18.2
19
19
17.8
17.8
18.984
1.05
20
20.2
21
19.4
20.6
21
19.8
19.4
20.2
20.2
21
19.4
21
20.6
21.4
19
20.2
21
20.6
18.6
19.4
18.2
19
19.4
19
19.8
20.2
19.8
19.4
20.6
20.6
19.4
20.2
20.2
19.4
19.4
20.6
19.8
19.4
20.6
19.8
20.6
19.8
19
19.8
21
19.8
21.4
19
20.2
19.988
0.9
21
20.4
20.8
20.8
19.2
21.6
20.4
19.2
22.6
19.2
22.8
20.4
20.4
21.6
22.8
20
21.2
20
20
21.2
20
20
20.4
20.8
20.8
20.8
20
22.4
22
21.6
20.8
20
22.4
21.6
22.4
21.2
20.8
19.6
19.6
20.8
22
21.2
19.6
21.2
20
21.6
21.6
20.8
20
21.6
20.861
1.15
22
22.6
22.2
22.2
22.2
22.6
22.2
23
23
21
22.6
22.2
20.2
21.4
22.6
23
22.2
22.2
22.2
22.6
22.6
23.8
23
22.6
22.2
24.2
21.4
23.8
23.4
20.2
20.6
22.6
23
21.4
21.8
21.8
22.2
23.8
21.8
23
23
24.6
20.6
23.4
23.4
21.8
21.8
22.2
23.4
21.8
22.396
1.02
23 Ave.
Range
22.8
21.08
3.4
23.2
21.16
4.6
22.4
21.08
3
24
20.68
6.6
23.2
22.04
2.2
23.6
21.32
3.8
22.8
20.68
4
22.8
21.6
3.6
21.2
20.2
2
21.2
21.16
4.6
21.6
20.44
3.6
25.2
21.08
6.6
22
21
2.6
22.4
21.88
2.6
22.8
20.36
6
24.4
21.56
4.6
23.6
21.32
3.8
22.4
20.76
3.8
22
20.52
4.4
23.2
20.76
4.6
22.4
20.6
5.6
24
21.08
5
22.8
20.68
5
23.2
20.6
5.4
22.8
21.32
5.2
21.6
20.52
2.2
22.8
21.64
4.4
23.2
21.4
4.4
20.8
20.36
3
23.2
20.84
4.2
24
20.76
6.2
23.2
21.8
3
20.4
20.92
1.4
22.8
20.92
4.6
22
20.68
3
24
21.4
4.6
22
20.68
5.6
23.2
20.44
5
22
20.76
5.6
24.4
21.96
4.6
25.6
22.44
5.4
25.6
20.76
7.4
22
21
4.4
23.6
21.32
3.8
23.2
21.16
5
24
21.24
5
23.6
21.4
4.6
23.6
20.76
5.8
23.2
20.92
5.4
22.980
21.042
1.17
1.058
Data
Set
38
39
40
41
42
43
44
45
46
Station
19
18.2
18.2
17.4
20.6
20.2
18.2
19.4
19.8
18.2
20
19.8
19.4
20.6
19.8
20.6
19.8
19
19.8
21
21
19.6
19.6
20.8
22
21.2
19.6
21.2
20
21.6
22
23.8
21.8
23
23
24.6
20.6
23.4
23.4
21.8
23 Ave. Range
22 20.68
5.6
23.2 20.44
5
22 20.76
5.6
24.4 21.96
4.6
25.6 22.44
5.4
25.6 20.76
7.4
22
21
4.4
23.6 21.32
3.8
23.2 21.16
5
Note: Sample 42 – All Values above mean with two by more
Than 2 std. Dev.
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
Summary
A
UCL
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Proposed Screen
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
Parameters Required to Calculate Control
Limits for the Proposed Charts
• Within Station Standard Deviation Inherent in the
Process
• Position Allowance for Maximum Position
• Position Allowance for Minimum Position
Estimation of Within Position Inherent
Standard Deviation
• Estimate from Within Position Moving
Range Data
• Estimate from Analysis of Variance
Residual after Removing Effects of Time
and Position
• Estimate from Analysis of Sample Means
• Compare to Historical Data
Estimates of Standard Deviation Based on Within
Positon Moving Range
Estimate of Standard Deviation based on Range
Parameter
19
20
21
22
Average
18.984 19.988 20.861 22.396
R-bar MR(2) 1.05
0.9
1.15
1.02
d2
1.128
1.128
1.128
1.128
Est. Sigma 0.931
0.798
1.020
0.904
23 Ave.
22.980 21.042
1.17
1.058
1.128
1.128
1.037
0.938
Analysis of Variance for Values - Type III Sums of Squares
-------------------------------------------------------------------------------Source
Sum of Squares
Df
Mean Square
F-Ratio
P-Value
-------------------------------------------------------------------------------MAIN EFFECTS
A:Set
56.5433
48
1.17799
1.34
0.0868
B:Station
537.441
4
134.36
152.80
0.0000
RESIDUAL
168.831
192
0.87933
-------------------------------------------------------------------------------TOTAL (CORRECTED)
762.815
244
-------------------------------------------------------------------------------All F-ratios are based on the residual mean square error.
Since Time is not significant, the SS for Time and
Error can be pooled to improve the estimate of s.
Factor
Error
Time
Pooled Error
SS
df MS
s
168.831 192
0.87933 0.938
56.5433 48
225.3743 240 0.939059583 0.969
Moving Range Chart for Sample Averages
M oving Range Chart
Summary
Moving Range
2.0
1.888
1.5
1.0
0.578
0.5
0.000
0.0
10
20
30
40
50
Sep 27, 2002 11:08:54
Estimate of Standard Deviation Based on Analysis of
Sample Averages
R-bar - Moving Rande Set Averages
d2
sy-bar
y-bar
UCLy-bar
LCLy-bar
Sample Size
s
0.578
1.128
0.512
21.043
22.580
19.506
5.000
1.146
Individuals Control Chart of Sample Averages
Ite m Chart
Summary
23
22.580
MV.Mean
22
21.042
21
20
19.504
19
10
20
30
40
Oct 6, 2002 16:39:38
Estimation of Position Effects PMax & PMin
• Historical Position Averages when Process is
Stable
• Analysis of Variance – Position Means
• Engineering Judgment of Reasonable Ranges
Mean = 21.0  = 1 Pmin = Pmax = 2
Chart Parameters
Parameter
Calculation
Value
Center Line
Y-double bar
21.00
UCL Average Y-double bar +
3*/?5
22.34
LCL Average
19.66
Y-double bar 3*/?5
UCL Individual Y-double bar
Pmax + 3*
26
LCL Individual
16
Y-double
bar+Pmin - 3*
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
Summary
A
UCL
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Data from “In Control” Process
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
Summary
A
A
UCL
Out of Control
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Average for All Stations Increased by 1 for the Last Point
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
A
A
Summary
A
UCL
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Average for All Stations Increased by 1 for the Last 10 Points
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
Summary
A
UCL
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Std. Dev. For Station 20 Increased to 2 for last 5 points
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
A
Summary
UCL
CL
LCL
B
10
20
30
Summary
M in. & M ax. Values
USL
HFI Out of Spec. Low
CLS
BLSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Std. Dev. For Station 20 Increased to 2 for last 23 points
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
A
Summary
UCL
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Average for Station 21 Increased to 22 for last 10 points
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
A
Summary
UCL
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Average for Station 21 Increased to 22 for last 24 points
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
A
Summary
A
UCL
Out of Control
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Average for Station 21 Increased by 3 for last 6 points
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
27
25
23
21
19
17
15
23.0
22.5
22.0
21.5
21.0
20.5
20.0
19.5
19.0
Lid Holes Demonstration Data
A
A
Summary
A
UCL
Out of Control
CL
LCL
10
20
Summary
M in. & M ax. Values
30
USL
CLS
LSL
4 8 12 16 20 24 28 32 36 40 44 48
Values
Ind.
Average
Average for Station 21 Increased by 3 for last 24 points
27
25
23
21
19
17
15
40
Box Plot by M old (Station)
19
20
21
22
23
17.0
A
A
Average
16.5
A
AA
16.0
Summary
A
UCL
CL
15.5
15.0 B
14.5
18.0
17.5
17.0
16.5
16.0
15.5
15.0
14.5
14.0
We igtht
A
A
LCL
B
10
Summary
M in. & M ax. Values
A A A AA
AA
4 8 12 16 20 24 28 32
20
18.0
17.5
17.0
USL 16.5
16.0
CLS 15.5
15.0
LSL 14.5
14.0
30
Box Plot by M old (Station)
weight
Ind.
“Real World” Chart
1
2
3
4
5
6
19.5
19.0
18.5
18.0
17.5
17.0
16.5
16.0
AA
A
A
A
We ight
A
A
AA
Out
A of Control
A
A
A
10
20
20
Summary
M in. & M ax. Values
AAA
A A
USL
18
CLS
LSL
16
14
4 8 12 16 20 24 28 32
20
BaseWt
Ind.
Average
Data Through Set 35
A
AA
Summary
AAA A
A
AA
A A
UCL
CL
LCL
30
Box Plot by M old (Station)
18
16
14
1 2 3 4 5 6 7 8 9 10
19.5
19.0
18.5
18.0
17.5
17.0
16.5
16.0
We ight
AA A A
A
A
Out
A A AA
A A
A
A
B
10
BB
B
20
20
Summary
M in.
AAA & M ax. Values
USL
18
CLS
16
14
Summary
of Control
LSL
BBBBB BBBBBBBB BBBBB
B
B
B
4 8 12 16 20 24 28 32 36 40 44 48
B BB
B
30
20
BaseWt
Ind.
Average
Last 48 Data Sets
A
B
B
A
A UCL
CL
LCL
BB
40
Box Plot by M old (Station)
18
16
14
1 2 3 4 5 6 7 8 9 10
Average
Fill
1200.0
1199.5
1199.0
1198.5
1198.0
1197.5
1197.0
1196.5
1196.0
1206
1204
1202
1200
1198
1196
1194
1192
1190
A
Out
A of Control
A
Summary
A
A
A
A
A
UCL
CL
B
B
B
10
A
Summary
M in. & M ax. Values
4 8 12 16 20 24 28 32 36 40
LCL
B
20
30
40
Box Plot by M old (Station)
1206
USL 1204
1202
1200
CLS 1198
1196
1194
LSL 1192
1190 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
1 11 1 1 11 1 1 12 2 2 22
Fill
Ind.
24 Station Machine
Average
1197
1196
1195
1194
1193
1192
1191
1190
1189
Fill
A
Out of Control
A
A
UCL
CL
LCL
B
1
2
3
4
5
Summary
M in. & M ax. Values
A
1201
1197
1193
6
USL
7
1201
8
9
B
10
11
12
Box Plot by M old (Station)
1197
CLS 1193
1189
1185
Summary
Fill
Ind.
24 Station Rotary Machine
LSL
4
8
12
1189
1185 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
1 11 1 1 11 1 1 12 2 2 22
Evaluation of Screen Change
•
•
•
•
•
Robust Container
Relatively Low Production Rate
Stable Process with Minimal Problems
Before ~ 3300 Observation
After 1 Year ~ 3300 Observations
Comparison of Probability Distributions Section B Before and After
Probability Plot
Normal Probability
After
SectionB
Theoretica
.99
.95
.90
.75
.50
.25
.10
.05
.01
10.0
Test for Normality:
Not applicable
Before
SectionB
Theoretica
12.5
15.0
Se ctionB
17.5
20.0
Comparison of Frequency Histograms Section B
Before
Frequency
400
300
200
100
0
10.0
12.5
15.0
17.5
20.0
Frequency
After
600
500
400
300
200
100
0
10.0
12.5
15.0
17.5
20.0
Comparison of Statistics for Section B
Statistic
Before
After
Average
15.98
16.003
Q3
16.65
16.5
Q1
15.4
15.55
Q3-Q1 Range
1.25
0.95
Std. Dev. (Not 0.943
Normal)
0.749
Cpk
Normal)
1.003
(Not 0.787
Comparison of Probability Distributions Section A
Probability Plot
Normal Probability
After
SectionA
Theoretica
.99
.95
.90
.75
.50
.25
.10
.05
.01
Before
SectiionA
Theoretica
9
10
11
Se
Se ctiionA
ctionA
12
13
Comparison of Frequency Histograms Section A
Before
Frequency
400
300
200
100
0
9
10
11
12
13
After
Frequency
800
600
400
200
0
9
10
11
12
13
Comparison of Statistics for Section A
Statistic
Before
After
Average
11.013
11.232
Q3
11.25
11.35
Q1
10.75
11.1
Q3-Q1 Range
0.5
0.25
Std. Dev. (Not 0.393
Normal)
0.232
Cpk
Normal)
1.123
(Not 0.477
Compare Probability Distributions Height
Probability Plot
Normal Probability
Before
height
Theoretical
.99
.95
.90
.75
Test for Normality:
Not applicable
.50
.25
.10
.05
.01
9.800
After
Height
Theoretical
9.825
9.850
9.875
Height
he ight
9.900
9.925
Frequency
600
500 Before
400
300
200
100
0
9.800 9.825 9.850 9.875 9.900 9.925
Frequency
600
500 After
400
300
200
100
0
9.800 9.825 9.850 9.875 9.900 9.925
Comparison of Statistics for Height
Statistic
Before
After
Average
9.888
9.872
Q3
9.896
9.879
Q1
9.881
9.865
Q3-Q1 Range
0.015
0.014
Std. Dev. (Not 0.00118
Normal)
0.01169
Cpk
Normal)
1.208
(Not 0.907
Conclusion
• Process control has improved substantially
since the new screen was introduced on this
line.
• Since there is no control, it is not possible to
determine how much if any of the improvement
was due to the change.
• “Hawthorne” Effect
Other Areas to Consider
• Add Hidden Tests to Determine when a Change
Occurs Between or Within Stations. Display
Message when an “Out of Control” Condition
Occurs
• Replace Capability Index with and Index of
Potential Process Improvement
• Statistics for Measurement and Control of
Contaminates in Post-Consumer Flake
Process & Product Analysis
Richard E. Clark
(630) 584 0566
[email protected]