Transcript Six Sigma Class

1

Will help you gain knowledge in:
◦ Improving performance
characteristics
◦ Reducing costs
◦ Understand regression analysis
◦ Understand relationships between
variables
◦ Understand correlation
◦ Understand how to optimize
processes

So you can:
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◦
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Recognize opportunities
Understand terminology
Know when to get help
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Data
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32
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27
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30
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38
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33
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45
41
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15
8
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24
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18
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Fuel Economy of 50 automobiles (in mpg)
Plot a histogram and calculate the average and
standard deviation
Fuel Economy
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X  22.88
S  7.7266
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Number of Cars
12
10
8
6
4
2
0
0 to <6
6 to <12
12 to <18 18 to <24 24 to <30 30 to <36 36 to <42 42 to <48 48 to <54 54 to <=60
mgp
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MANPOWER
METHOD
MACHINE
MEASUREMENT
MOTHER
NATURE
MATERIAL
Experimental design (a.k.a. DOE)
is about discovering and
quantifying the magnitude of
cause and effect relationships.
We need DOE because intuition
can be misleading.... but we’ll
get to that in a minute.
Regression can be used to
explain how we can model data
experimentally.
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Let’s take a look at the mileage
data and see if there’s a factor
that might explain some of the
variation.
Draw a scatter diagram for the
following data:
X Y=f(X) Y
Observation
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X - Weight (lbs)
3000
2800
2100
2900
2400
3300
2700
3500
2500
3200
Y - Mileage(mpg)
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14
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12
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If you draw a best fit line and
figure out an equation for that
line, you would have a ‘model’
that represents the data.
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Y=f(X)
mpg
Scatter Chart (Weight vs mpg)
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30
25
20
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10
5
0
1900
y = -0.0152x + 63.507
2
R = 0.9191
2400
2900
3400
3900
Weight
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There are basically three ways to
understand a process you are working
on.
 Classical 1FAT experiments
◦ One factor at a time (1FAT) focuses on one
variable at two or three levels and attempts
to hold everything else constant (which is
impossible to do in a complicated process).
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Mathematical model
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DOE
◦ Express the system with a mathematical
equation.
◦ When properly constructed, it can focus on
a wide range of key input factors and will
determine the optimum levels of each of
the factors.
Each have their advantages and
disadvantages. We’ll talk about each.
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Let’s consider how two known
(based on years of experience)
factors affect gas mileage, tire
size (T) and fuel type (F).
T(1,2)
Y
Y=f(X)
F(1,2)
Fuel Type
Tire size
F1
T1
F2
T2
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Step 1:
Select two levels of tire size
and two kinds of fuels.
Step 2:
Holding fuel type constant
(and everything else), test the car
at both tire sizes.
Fuel Type Tire size
Mpg
F1
T1
20
F1
T2
30
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Since we want to maximize mpg
the more desirable response
happened with T2
Step 3: Holding tire size at T2, test
the car at both fuel types.
Fuel Type Tire size
Mpg
F1
T2
30
F2
T2
40
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Looks like the ideal setting is F2
and T2 at 40mpg.
This is a common experimental
method.
What about the possible
interaction effect of tire size and
fuel type. F2T1
Fuel Type Tire size
Mpg
F1
T2
30
F2
T2
40
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70
Suppose that the untested
combination F2T1 would produce
the results below.
There is a different slope so
there appears to be an
interaction. A more appropriate
design would be to test all four
combinations.
60
F2
mpg
50
40
30
F1
20
10
0
T1
T2
Tire Size
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We need a way to
◦ investigate the relationship(s) between variables
◦ distinguish the effects of variables from each other (and
maybe tell if they interact with each other)
◦ quantify the effects...
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...So we can predict, control, and optimize
processes.
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We can see some problems with
1FAT. Now let’s go back and
talk about the statapult.
We can do a mathematical
model or we could do a DOE.
DOE will build a ‘model’ - a
mathematical representation
of the behavior of
measurements.
or…
You could build a
“mathematical model” without
DOE and it might look
something like...
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DOE uses purposeful changes of the inputs
(factors) in order to observe corresponding
changes to the outputs (response).
Remember the IPO’s we did – they are real
important here.
Run
X1
X2
X3
X4
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+
+
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+
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+
+
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Y1
Y2
Y3
Y-bar
SY
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To ‘design’ an experiment,
means to pick the points that
you’ll use for a scatter diagram.
A
B
(-,+)
(+,+)
Factor B Settings
High (+)
X1
Y
X2
In tabular
form, it would
look like:
Run
A
B
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2
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4
+
+
+
+
Low (-)
(-,-)
Low (-)
(+,-)
Factor A Settings
High (+)
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Set objectives (Charter)
◦ Comparative
 Determine what factor is significant
◦ Screening
 Determine what factors will be studied
◦ Model – response surface method
 Determine interactions and optimize
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Select process variables (C&E) and
levels you will test at
Select an experimental design
Execute the design
CONFIRM the model!! Check that
the data are consistent with the
experimental assumptions
Analyze and interpret the results
Use/present the results
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http://jimakers.com/downloads/DOE_Setup.docx
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A full factorial is an experimental
design which contains all levels
of all factors. No possible
treatments are omitted.
◦ The preferred (ultimate) design
◦ Best for modeling
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A fractional factorial is a
balanced experimental design
which contains fewer than all
combinations of all levels of all
factors.
◦ The preferred design when a full
factorial cannot be performed due to
lack of resources
◦ Okay for some modeling
◦ Good for screening
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Full factorial
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2 level
3 factors
8 runs
Balanced
(orthogonal)
23  8 runs
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231  4 runs
Fractional
factorial
◦ 2 level
◦ 3 factors
◦ 4 runs - Half
fraction
◦ Balanced
(orthogonal)
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Response - Y
Average Y
when A was
set ‘high’
Average Y
when A was
set ‘low’
Low
High
Factor A
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The difference in the average Y when
A was ‘high’ from the average Y when
A was ‘low’ is the ‘factor effect’
The differences are calculated for
every factor in the experiment
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B = High
Response - Y
Response - Y
When the effect of one factor changes
due to the effect of another factor, the
two factors are said to ‘interact.’
B = Low
Strong
B = Low
B = High
Low
Slight
High
Response - Y
Factor A
B = Low
None
B = High
Low
High
Low
High
Factor A
more than two
factors can interact
at the same time,
but it is thought to
be rare outside of
chemical reactions.
Factor A
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Using the statapult, we will
experiment with some factors to
“model” the process.
We will perform a confirmation run to
determine if the model will help us
predict the proper settings required
to achieve a desired output.
A
B
C
D
X1
X2
X3
X4
Y
Y=f(X1, X2, X3, X4)
What design should we use?
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Too much variation in the
response
Measurement error
Poor experimental discipline
Aliases (confounded) effects
Inadequate model
Something changed
- And: -
There may not be a true
cause-and-effect relationship. 
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2  16 run
4
Full FactorialExperiment
Factor
A
B
C
D
Response #1
Row #
A-
B-
C-
D-
Y1
1
-1
-1
-1
-1
2
-1
-1
-1
1
3
-1
-1
1
-1
4
-1
-1
1
1
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-1
1
-1
-1
6
-1
1
-1
1
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-1
1
1
-1
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-1
1
1
1
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1
-1
-1
-1
10
1
-1
-1
1
11
1
-1
1
-1
12
1
-1
1
1
13
1
1
-1
-1
14
1
1
-1
1
15
1
1
1
-1
16
1
1
1
1
Y2
Y3
Confirmation runs
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Full factorial
3 level
3 factors
27 runs
Balanced
(orthogonal)
◦ Used when it is expected the
response in non-linear
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33  27 runs
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Useful to see how factors effect the
response and to determine what
other settings provide the same
response
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Helpful in reaching the optimal
result
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