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2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)
Spring 2008
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Control of
Manufacturing Processes
Subject 2.830/6.780/ESD.63
Spring 2008
Lecture #4
Probability Models of
Manufacturing Processes
February 14, 2008
Manufacturing
1
Note: Reading Assignment
• May & Spanos
– Read Chapter 4
• Montgomery
– Skim/consult Chapters 2 & 3 if need additional
explanations or examples beyond May & Spanos
Manufacturing
2
Turning Process
Manufacturing
3
Observations from Experiments
• Randomness + Deterministic Changes
CNC Turning
0.7020
random or
unknown
0.7010
Δα
0.7000
0.6990
0.6980
Shift changes
0.6970
Run Number
Manufacturing
4
CNC Data
0.6253
Outer
Middle
Inner
0.6251
0.6249
Operating Points:
High Feed=1
Low Feed = 2
0.6247
0.6245
0.6243
0.6241
0.6239
0.6237
0.6235
1
2
3
Manufacturing
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
5
Brake Bending of Sheet
Output:
Angle
M
M
ε
σ
Manufacturing
6
Bending Process
Springback
Manufacturing
7
Observations from Bending Process
• Clear Input-Output Effects (Deterministic)
• Also Randomness as well
90
80
Steel 0.6 In
70
Aluminum 0.6 In
60
Steel 0.3 In
50
40
Aluminum 0.3 In
Angle
changes
with depth
ΔY ⇒ Δu
30
20
Manufacturing
8
Observations from Injection Molding
QuickTime™ and a
Graphics decompressor
are needed to see this picture.
Run Chart for Injection Molded Part
Holding Time = 5 sec
Injection Press = 60%
41.00
Holding Time = 5 sec
Injection Press = 40%
40.95
Holding Time = 10 sec
Injection Press = 40%
Holding Time = 10 sec
Injection Press = 60%
2/3/05
50
40.90
40.85
Width (mm)
Average
40.80
40.75
40.70
40.65
40.60
0
10
20
30
40
50
60
Number of Run
Manufacturing
9
Observations from Data
• Clearly some measurement “noise”?
90
80
70
60
shift 3
50
40
shift 1
shift 2
30
20
Manufacturing
10
Observations from Data
• Systematic/traceable “operator error”
Sheet Shearing
1.01
0.99
0.97
0.95
steel
0.93
0.91
0.89
0.87
shift
change
steel
shift
change
aluminum
0.85
Manufacturing
11
How Model to Distinguish these Effects?
Irreducible
Disturbances
Inputs
Process
Outputs
+
"Noise"
Disturbances
(Reducible)
A Random Process + A Deterministic Process
Manufacturing
12
Random Processes
• Consider the Output-only, “Black Box” view of the
Run Chart
Process
Y(t)
• How do we characterize the process?
– Using Y(t) only
• WHY do we characterize the process
– Using Y(t) only?
Manufacturing
13
The Why
•
•
•
•
Did output really change?
Did the input cause the change?
If not, why did the output vary?
How confident are we of these answers?
Process
• Can we model the randomness?
Manufacturing
14
Background Needed
• Theory of Random Processes and Random
Variables
• Use of Sample Statistics Based on
Measurements
– SPC basis
– DOE: use of experimental I/O data
– Feedback control with random disturbances
Manufacturing
15
How to Describe Randomness?
• Look at a Frequency Histogram of the Data
• Estimates likelihood of certain ranges
occurring:
– Pr(y1 < Y <y2)
– “Probability that a random variable Y falls
between the limits y1 and y2”
Manufacturing
16
Example: Thermoforming Histogram (2000 data)
12
1.505
Thermoforming Run Chart
1.5
1.495
10
1.49
Part Diameter
Shift Changes
1.485
1.48
8
1.475
Run Number
6
4
2
0
1.48
1.482
Manufacturing
1.484
1.486
1.488
1.49
1.492
1.494
1.496
1.498
1.5
1.502
17
How to Describe Continuous Randomness
• Process outputs Y are continuous variables
• The Probability of Y(t) taking on any specific value for a
continuum
Prob(Y(t) = y*) = 0
• Must use instead a Cumulative Probability Function
Pr(Y(t)<y*)
– Look at Cumulative Frequency
Manufacturing
18
Cumulative Frequency
60
12
10
8
6
50
4
2
0
1.48
1.482
1.484
1.486
1.488
1.49
1.492
1.494
1.496
1.498
1.5
1.502
A Continuous equivalent?
40
Pr(Y<1.49)=14/50
= 0.28
30
20
10
0
1.48
1.482
Manufacturing
1.484
1.486
1.488
1.49
1.492
1.494
1.496
1.498
191.5
Continuous Equivalents
• Probability Function: (P(x))
60
P
50
40
30
20
10
x
• Probability Density Function pdf(x) = dP/dx
0
1.48
1.482
1.484
1.486
1.484
1.486
1.488
1.49
1.492
1.49
1.492
1.494
1.496
1.498
1.5
12
p(x)
10
8
6
4
2
0
x
Manufacturing
1.48
1.482
1.488
1.494
1.496
1.498
1.5
1.502
20
Process Outputs as a Random Variable
• The Histogram suggests a pdf
– Parent or underlying behavior “sampled” by the
process
• Standard Forms (There are Many)
– e.g. The Uniform and Normal pdf’s
Normal Probability Density Function for σ2 = 1, μ=0
0.4
p(x)
0.3
0.2
0.1
-4
-3
-2
-1
0
1
2
3
4
x
Manufacturing
21
Analysis of Histograms
• Is there a consistent pattern?
• Is an underlying “parent” distribution suggested?
Manufacturing
22
Histogram for CNC Turning
CNC Turning
0.7020
25
0.7010
0.7000
0.6990
20
15
10
5
0.6980
0
0.697
0.698
Shift changes
0.7
0.699
0.6970
0.701
0.702
47
45
43
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
3
1
Bin Dimension (in)
Run Number
Manufacturing
23
Histogram for Bending
(MIT 2002 data)
Bending Data Sorted by Depth
150
140
14
0.3 in depth only
Dashed Lines are at group changes
120
12
110
Steel
0.6 in
depth
Aluminum
0.6 in depth
100
Aluminum
0.3in depth
10
Steel
0.3 in
depth
90
133
129
125
121
117
113
109
105
97
101
93
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
8
17
9
13
5
80
1
Angle (Deg.)
130
6
4
2
0
134
Manufacturing
134.5
135
135.5
136
136.5
137
137.5
138
138.5
139
139.5
140
140.5
141
141.5
142
24
Histogram for Bending
(MIT 2002 data)
Bending Data Sorted by Depth
150
0.6 in depth only
140
Dashed Lines are at group changes
18
Angle (Deg.)
130
120
110
16
Steel
0.6 in
depth
Aluminum
0.6 in depth
100
Aluminum
0.3in depth
Steel
0.3 in
depth
90
14
133
129
125
121
117
113
109
97
105
93
101
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
9
13
5
1
80
12
10
8
6
4
2
0
98
98.5
99
99.5
100
100.5
101
101.5
102
102.5
103
103.5
104
104.5
Mean Shift?
Manufacturing
25
Consider: No Intentional Changes (Δu = 0)
• Shearing during shift 1(‘02) , aluminum only
60
50
40
30
20
10
0
0.91
Manufacturing
0.915
0.92
0.925
0.93
0.935
0.94
0.945
0.95
26
Consider: No Effective Changes (∂Y/∂u= 0)
• Injection Molding Entire MIT Run (2002)
25
1.895
1.89
20
1.885
1.88
1.875
15
Shift
Change
1.87
1.865
10
5
0
1.875
Manufacturing
1.8775
1.88
1.8825
1.885
1.8875
1.89
27
Injection Molding (S’2003)
66.4
40
66.2
35
66
65.8
Frequency
30
65.6
65.4
25
65.2
1
8
15
22
29
36
43
50
57
64
71
78
85
92
99 106 113 120 127
20
15
10
5
0
65.4 65.5 65.5 65.6 65.6 65.7 65.7 65.8 65.8 65.9 65.9
Manufacturing
66
66
28
Conclusion?
• When there are no input effect (no Δu or ∂Y/∂u) a
consistent histogram pattern can emerge
• How do we use knowledge of this pattern?
– Predict behavior
– Set limits on “normal” behavior
• Define analytical probability density functions
Manufacturing
29
Underlying or “Parent” Probability
• A model of the “true”, continuous behavior of the
random process
• Can be thought of as a continuous random variable
obeying a set of rules (the probability function)
• We can only glimpse into these rules by sampling the
random variable (i.e. the process output)
• Underlying process can have
– Continuous values (e.g. geometry)
– Discrete values (e.g. defect occurrence)
Manufacturing
30
Continuous Probability Functions
• Recall Probability Function (Cumulative)
P(x) = Prob(Y(t)<x)
P
• Define pdf = p(x) = dP/dx
Thus :
P(x) =
∫
x
−∞
x
p(x)
pdf (x)dx
x
Manufacturing
31
Use of the pdf : Expectation
E{x(t)} = expected value of x(t)
E{x(t)} =
∫
∞
-∞
x(t) pdf (x,t )dx
μ(t) = E{x(t)} mean value of x
Note that pdf and μ (or any other expected value)
can be functions of time.
In general, they may be non-stationary.
Manufacturing
32
Stationary Processes
pdf(x,t) = pdf(x) = p(x) : stationary pdf
E{x} =
∞
∫x
p(x)dx = μx
−∞
μx : theoretical or " true" mean
• For a stationary process μx is a constant
Manufacturing
33
Stationary Processes
E{( x − μx ) } =
2
∞
∫ (x − μ )
2
x
p(x)dx = σ
2
x
−∞
= "true" variance
• For a stationary process σx is a constant
σ = E{x } − μ
2
x
2
2
x
= mean square - square of mean
Manufacturing
34
The Uniform Distribution
1
r
p(x) = 0
p(x) =
p(x)
⇒ x1 < x < x 2
⇒ x < x1
x > x2
1/r
x1
Manufacturing
r
x2
x
35
The Normal Distribution
1
p(x) =
e
σ 2π
1 ⎛ x− μ⎞
− ⎜
⎟
2⎝ σ ⎠
2
0.4
“Standard normal”
z=
0.3
x−μ
σ
0.2
μz=0
σz=1
0.1
0
-4
-3
-2
-1
0
1
2
3
4
z
Manufacturing
36
Cumulative Distribution
P(z)
1
0.8
0.6
0.4
0.2
0
-4
-4
-3
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
3
4
4
z
Manufacturing
37
Properties of the Normal pdf
• Symmetric about mean
• Only two parameters:
μ and σ
1
p(x) =
e
σ 2π
1 ⎛ x−μ⎞
− ⎜
⎟
2⎝ σ ⎠
2
• Superposition Applies:
– sum of normal random variables has a normal
distribution
Manufacturing
38
Superposition of Random Variables
If we define a variable
y = c1x1 + c2x2 + c3x3 + c4x4 + ...
• ci are constants
• xi are independent random variables
μy = c1μ1 + c2μ2 + c3μ3 + c4μ4 + ...
σy2 = c12σι2 + c22σ22 + c32σ32 + c42σ42
From expectation operation, for any pdf.
Manufacturing
39
Use of the PDF: Confidence Intervals
• How likely are certain values of the random
variable?
• For a “Standard Normal” Distribution:
z=
(x − μ )
σ
N(0,1)
μ=0
σ =1
z = 1 ⇒ x = 1σ
z = 2 ⇒ x = 2σ
z = 3 ⇒ x = 3σ
0.4
0.3
0.2
0.1
-4
Manufacturing
-3
-2
-1
0
0
1
2
3
4
40
Confidence Intervals
P(-1≤ z ≤ 1) = P(z ≤1) - P(z ≤-1) = 0.841 - (1-0.841) = 0.682
(+ 1σ)
P(-2≤ z ≤ 2) = P(z ≤2) - P(z ≤-2) = 0.977 - (1-0.977) = 0.954
(+ 2σ)
P(-3≤ z ≤ 3) = P(z ≤3) - P(z ≤-3) = 0.998 - (1-0.998) = 0.997
(+ 3σ)
P(z) tabulated (e.g. p. 752 of Montgomery)
Manufacturing
41
Confidence Intervals
0.4
0.3σ = 3/1000
• Probability that |x| >μ+3
68%
0.2
95%
0.1
99.7%
-4
-3
-2
-1
0
1
2
3
4
z
±σ
±2 σ
±3 σ
Manufacturing
42
Is the Process “Normal” ?
• Is the underlying distribution really normal?
– Look at histogram
– Look at curve fit to histogram
– Look at % of data in 1, 2 and 3 σ bands
• Confidence Intervals
– Probability (or qq) plots (see Mont. 3-3.7)
– Look at “kurtosis”
• Measure of deviation from normal
Manufacturing
43
Kurtosis: Deviation from Normal
k=
E( x − μ x )
σ
4
4
k=1 - normal
k>1 more “peaked”
k<1 more “flat”
Or for sampled data:
4
⎡
n(n + 1)
3(n − 1)2
⎛ xi − x ⎞ ⎤
k=⎢
⎟ ⎥−
⎜
∑
⎢⎣ (n − 1)(n − 2)(n − 3) ⎝ s ⎠ ⎥⎦ (n − 2)(n − 3)
Manufacturing
44
The Central Limit Theorem
– If x1, x2 ,x3 ...xN … are N independent observations of
a random variable with “moments” μx and σ2x,
– The distribution of the sum of all the samples will
tend toward normal.
Manufacturing
45
Example: Uniformly Distributed Data
70
60
x1
50
40
30
100
y = ∑ xi
20
10
0
Sum of 100 sets of
1000 points each
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
+
i =1
150
70
60
50
x2
40
30
20
100
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
+...
50
70
60
50
x100
40
30
20
0
35
40
45
50
55
60
65
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Manufacturing
0.7
0.8
0.9
1
46
Sampling: Using Measurements (Data)
to Model the Random Process
•
•
In general p(x) is unknown
Data can suggest form of p(x)
– e.g.. uniform, normal, weibull, etc.
•
Data can be used to estimate parameters of distributions
– e.g. μ and σ for normal distribution - p(x) = p(x, μ ,σ)
•
How to Estimate
– Sample Statistics
•
Uncertainty in Estimates
1
p(x) =
e
σ 2π
1 ⎛ x− μ⎞
− ⎜
⎟
2⎝ σ ⎠
2
– Sample Statistic pdf’s
Manufacturing
47
Sample Statistics
x( j) = samples of x(t ) taken n times
1 n
x = ∑ x( j) : Average or Sample Mean
n j =1
n
1
2
2
S =
(x( j) − x ) : Sample Variance
∑
n − 1 j =1
S =
1 n
2
(x( j) − x ) : Sample Std.Dev.
∑
n − 1 j =1
Manufacturing
48
Sample Mean Uncertainty
• If all xi come from a distribution with μx and σ2x, and
we divide the sum by n:
1 n
x = ∑ xi
n i =1
Then:
x = c1 x1 + c2 x2 + c3 x 3 + K cn xn
1
ci =
n
μx = μx
Manufacturing
and
1 2
1
σ = σ x or σ x =
σx
n
n
2
x
49
Conclusions
• All Physical Processes Have a Degree of
Natural Randomness
• We can Model this Behavior using Probability
Distribution Functions
• We can Calibrate and Evaluate the Quality of
this Model from Measurement Data
Manufacturing
50