Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Taguchi Robust Designs

/3805 BB W4 Taguchi 07, D. Szemkus/H. Winkler
Taguchi
Robust Designs
Taguchi's Quadratic Loss Function
0
50
100
150
200
200 300 400 500 600
Output Response
Loss
Week 4
Page 2/3805 BB W4 Taguchi 07, D. Szemkus/H. Winkler
What do we mean if we talk about a robust
design or product?
Name examples about robust designs or products in
your environment?
An example for a continuous process?
... For a discrete process?
... In development?
... In administration?
Introduction

Goals of the Module
• Understanding of the principle of robust designs
within product and process development
• Understanding of the Taguchi method -
application of a DOE to investigate robustness
• Creation and analysis of experiments to
investigate factors which have the most influence
on a product / process design or variation
• Evaluation of interactions in order to make a
process robust
Early 1980 Prof. Genechi Taguchi applied the
statistical design of experiment:
− to design products or processes which are
robust against noise factors like raw materials,
employees, environment…
− to minimize variation around a target value
The key is the reduction of the variation
Taguchi Method

The Concept of the Loss Function - Taguchi
Quality can be presented as a deviation form target value
or the nominal value due to the following equation:
L(y) = k(y-T)2
Taguchi's Quadratic Loss Function
0
50
100
150
200
200 300 400 500 600
Output Response
Loss
Name an example of such a loss
function in your area!
Why increases the manufacturing
costs in your example at a
deviation form the target or a
nominal value?
The Concept of the Loss Function - Taguchi

Taguchi Philosophy
Products shall be robust against variation due to
external causes
In order to achieve robustness we can apply the
DOE technique as a product development tool
Processes „ON-TARGET“ are more important
than the fulfillment of specification limits
Taguchi Concept
• System Design
– Research and development does define the settings for
a process / product
• Parameter Design
– The accurate settings of the process / product
parameter (nominal-value) will be defined
• Tolerance Design
– The acceptable variation range for process / product
parameter (nominal-value) will be defined
• Controllable Parameter (controllable variables, Inner Array)
and uncontrollable parameter (noise variables, Outer Array)
will be defined

Parameter Design
We differentiate between four controllable
parameter / factors
Factors which influences the mean and the
standard deviation
Factors which influences the mean only
Factors which influences the standard
deviation only
Factor which influences neither the mean
and/or the standard deviation
Factors which Influences the Mean and σ
600500400300200
40000
30000
20000
10000
0
Factor
Response
Why is the variation at a higher factor level of the
output lower than at a low factor level?
Do we expect the same effect if we have a linear
relation?

Factors which Influences the Mean Only
600500400300200
Factor
Response
What is variation of the outputs in relation to the factor
level in this case?
We look for an interaction of a factor (e.g. product type) with the
noise variable of the customer application. Which product type is
more robust against changes of the noise variable?
600500400300200
Noise variable
Response
Product Type 1
Product Type 2
Factors which Influences the σ Only

Factors which Influences the σ Only
600500400300200
Noise variables
Response
changeable factors
We Use the Interaction
600500400300200
Setting of variable A
to a minimal
variation
Variation due to variable B
We look for an interaction of a controllable variable (e.g.
temperature) with a noise variable (e.g. impurity of the raw material)
of our process.

Taguchi uses the concept of the Inner / Outer Arrays to investigate
the effects of the controllable variables (inner array variables) on
the output of the process in dependency of the uncontrolled noise
variables (outer array variables).
The uncontrolled variables or noise variables can be related to the
customer process, the supplier process (raw material conditions) or
subsequent steps in our own processes.
What could be Taguchi’s reasons to develop this kind of concept?
What advantages but also disadvantages has this approach?
The concept of Inner and Outer Arrays
Taguchi Design in Minitab
Stat
>DOE
>Taguchi
>Create Taguchi Design …
Stat
>DOE
>Taguchi
>Create Taguchi Design …
Start with the definition of the controllable factors.

Comparison of a factorial and a Taguchi design
With 3 factors and 2 levels, the
Taguchi design is slightly
different than the standard
design.
The orthogonal character
sustains. Same experiments with
a changed order.
A factorial design can be
extended with the Taguchi
idea.
A B C StdOrder Taguchi Order
-1 -1 -1 1 1
1 -1 -1 2 5
-1 1 -1 3 3
1 1 -1 4 7
-1 -1 1 5 2
1 -1 1 6 6
-1 1 1 7 4
1 1 1 8 8
2k
factorial Design
A B C Taguchi Order StdOrder
1 1 1 1 1
1 1 2 5 2
1 2 1 3 3
1 2 2 7 4
2 1 1 2 5
2 1 2 6 6
2 2 1 4 7
2 2 2 8 8
Taguchi Design
A B C
1 1 1
1 2 2
1 3 3
2 1 2
2 2 3
2 3 1
3 1 3
3 2 1
3 3 2
A B C
1 1 1
1 1 1
1 1 1
1 2 2
1 2 2
1 2 2
1 3 3
1 3 3
1 3 3
2 1 2
2 1 2
2 1 2
2 2 3
2 2 3
2 2 3
2 3 1
2 3 1
2 3 1
3 1 3
3 1 3
3 1 3
3 2 1
3 2 1
3 2 1
3 3 2
3 3 2
3 3 2
Taguchi designs with 3 factors
and 3 level results in 9 or 27
experiments
With 9 runs the design has an fractional
factorial character, with 27 runs it
corresponds to a full factorial design.

1 1 2 2 A
1 2 1 2 B
A B C Y 1 Y 2 Y 3 Y 4
1 1 1
1 1 2
2 2 1
2 2 2
1 2 1
1 2 2
2 1 1
2 1 2
Inner Array
Outer Array
The concept of Inner and Outer Arrays
In this example we will run the experiments under 4 predefined condition.
The inner and the outer array have a full factorial character. The inner and
as well the outer array can be included in a fractional factorial design.
The noise factors have to be added manually. In the easiest way the runs
will be just repeated. You will receive more information if the noise factors
will be investigated over their adjustment range.
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
Σ∗−=
n
YSN
2
1
log10
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Σ∗−=
n
Y
SN
2
log10
Signal to Noise Ratio
In the evaluation of a Taguchi design the variation receives a special
emphasis due to the calculation of the Signal to Noise Ratio .
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∗= 2
2
log10
s
Y
SN
Nominal is best
Target the response and you want to base the
S/N ratio on means and standard deviations.
For advanced factorial designs you can calculate the SN ration manually
and therefore useful for others design as well.
The calculation of the Signal to Noise Ratio
Larger is better
Maximize the response
Smaller is better
Minimize the response

Example: Ceramic furnace
Problem: Differences in the size after baking process of ceramic
Datei: Ceramic Taguchi.mtw
The design includes 7
factors with 2 levels.
The noise factor is the
position in the furnace. The
ceramic size has been
measured at 5 position after
the treatment.
A B C D E F G P 1 P 2 P 3 P 4 P 5
old A 1 0 1200 43 0 168,30 167,20 166,43 165,00 164,45
old A 1 4 1300 53 5 169,95 168,63 166,98 165,44 164,56
old B 5 0 1200 53 5 172,15 167,31 165,33 163,35 158,73
old B 5 4 1300 43 0 166,65 165,88 165,66 165,22 164,67
new A 5 0 1300 43 5 167,42 166,43 166,21 165,66 165,00
new A 5 4 1200 53 0 168,41 166,98 166,98 166,54 165,66
new B 1 0 1300 53 0 166,65 165,88 165,00 164,34 164,01
new B 1 4 1200 43 5 167,09 166,54 165,44 165,22 164,56
Level 1 Level 2
A Material A old new
B Quality Additive B A B
C Portion % Additive B 5 1
D Recl.-Portion %. 0 4
E Filling Quantity kg 1300 1200
F Portion % Material C 43 53
G Portion % Material D 0 5
Factor
Stat
>DOE
>Taguchi
>Analyze Taguchi Design …
Stat
>DOE
>Taguchi
>Analyze Taguchi Design …
Terms
Analysis
Graphs

The mean differences are rated in the Session Window.
There are almost no differences of SN Ratio for the factors B and C.
The factor B is most capable to shift the mean.
Taguchi Analysis: P 1; P 2; P 3; P 4; P 5 versus A; B; C; D; E; F; G
Response Table for Signal to Noise Ratios
Nominal is best (10*Log(Ybar**2/s**2))
Level A B C D E F G
1 38,87 41,95 41,44 39,96 39,90 44,23 43,89
2 44,39 41,31 41,82 43,30 43,36 39,03 39,37
Delta 5,52 0,64 0,37 3,34 3,46 5,20 4,52
Rank 1 6 7 5 4 2 3
Response Table for Means
Level A B C D E F G
1 166,1 166,6 166,1 165,7 166,1 166,0 166,0
2 166,0 165,5 166,0 166,4 166,0 166,1 166,1
Delta 0,1 1,1 0,1 0,6 0,1 0,2 0,1
Rank 5 1 6,5 2 6,5 3 4
SN Ratio: Maximization
it means variation
reduction
MeanofMeans
neualt
166,5
166,0
165,5
BA 51
40
166,5
166,0
165,5
13001200 5343
50
166,5
166,0
165,5
A B C
D E F
G
Main Effects Plot (data means) for Means
MeanofSNratios
neualt
44
42
40
BA 51
40
44
42
40
13001200 5343
50
44
42
40
A B C
D E F
G
Main Effects Plot (data means) for SN ratios
Signal-to-noise: Nominal is best (10*Log(Ybar**2/s**2))

MeanofMeans
neualt
166,5
166,0
165,5
BA 51
40
166,5
166,0
165,5
13001200 5343
50
166,5
166,0
165,5
A B C
D E F
G
Main Effects Plot (data means) for Means
MeanofStandardDeviations neualt
2,0
1,5
1,0
BA 51
40
2,0
1,5
1,0
13001200 5343
50
2,0
1,5
1,0
A B C
D E F
G
Main Effects Plot (data means) for Standard Deviations
SN Ratio: Maximization
it means variation
reduction
Example: Cake Study
• Goal: A cake with a good taste
• Output: Taste scale (the higher the better)
• Controllable inputs ( Inner Array):
– Flour (low, recommended, high)
– Butter (low, recommended, high)
– Eggs (low, recommended, high)
• Uncontrollable inputs ( Outer Array):
– Backing temp (Low, recommended, high)
– Backing time (Low, recommended, high)

Example: Cake Study
0 -1 1 -1 1 Temp
0 -1 -1 1 1 time
Flour Butter Egg T0 t0 T-1 t-1 T+1 t-1 T-1 t+1 T+1 t+1
0 0 0 6,7 3,4 5,4 4,1 3,8
-1 -1 -1 3,1 1,1 5,7 6,4 1,3
1 -1 -1 3,2 3,8 4,9 4,3 1,3
-1 1 -1 5,3 3,7 5,1 6,7 2,9
1 1 -1 4,1 4,5 6,4 5,8 5,2
-1 -1 1 5,9 4,2 6,8 6,5 3,5
1 -1 1 6,9 5 6 5,9 5,7
-1 1 1 3 3,1 6,3 6,4 3
1 1 1 4,5 3,9 5,5 5 5,4
Kontrollierbare Faktoren Inner
Array
Störfaktoren (Noise) Outer Array
CAKE.MTWCAKE.MTW
Example: Cake Study
0 -1 1 -1 1 Temp
0 -1 -1 1 1 time
Flour Butter Egg T0 t0 T-1 t-1 T+1 t-1 T-1 t+1 T+1 t+1
0 0 0 6,7 3,4 5,4 4,1 3,8
-1 -1 -1 3,1 1,1 5,7 6,4 1,3
1 -1 -1 3,2 3,8 4,9 4,3 1,3
-1 1 -1 5,3 3,7 5,1 6,7 2,9
1 1 -1 4,1 4,5 6,4 5,8 5,2
-1 -1 1 5,9 4,2 6,8 6,5 3,5
1 -1 1 6,9 5 6 5,9 5,7
-1 1 1 3 3,1 6,3 6,4 3
1 1 1 4,5 3,9 5,5 5 5,4
Kontrollierbare Faktoren Inner
Array
Störfaktoren (Noise) Outer Array
CAKE.MTWCAKE.MTW
Stat
>DOE
>Factorial
>Define factorial design …
Stat
>DOE
>Factorial
>Define factorial design …

10-1
5,0
4,8
4,6
4,4
4,2
10-1
10-1
5,0
4,8
4,6
4,4
4,2
Flour
Mean
Butter
Egg
Corner
Center
Point Type
Main Effects Plot for Mean
Data Means
Main Effects
Center points are not significant (curvature)Center points are not significant (curvature)
Stat
>DOE
>Factorial
>Factorial Plots …
Stat
>DOE
>Factorial
10-1 10-1
5,6
4,8
4,0
5,6
4,8
4,0
Flour
Butter
Egg
-1 Corner
0 Center
1 Corner
Flour Point Type
-1 Corner
0 Center
1 Corner
Butter Point Type
Interaction Plot for Mean
Data Means
Interactions
The interaction between
butter and eggs is
significant
The interaction between
butter and eggs is
significant
Stat
>DOE
>Factorial
Stat
>DOE
>Factorial

Factorial Fit: Mean versus Flour; Butter; Egg
Estimated Effects and Coefficients for Mean (coded units)
Term Effect Coef SE Coef T P
Constant 4,6822 0,04167 112,36 0,000
Flour 0,3650 0,1825 0,04420 4,13 0,054
Butter 0,2150 0,1075 0,04420 2,43 0,136
Egg 0,8850 0,4425 0,04420 10,01 0,010
Flour*Butter 0,1150 0,0575 0,04420 1,30 0,323
Flour*Egg 0,1450 0,0725 0,04420 1,64 0,243
Butter*Egg -1,2450 -0,6225 0,04420 -14,08 0,005
Statistical Analysis of the Taste Rating
Enter column C9 for “mean” and select
“terms”. Include all terms except ABC
and center points in model.
Butter * Egg interaction is the most importantButter * Egg interaction is the most important
Stat
>DOE
>Analyze factorial design …
Stat
>DOE
Which factors (parameter) have a significant effect on the
taste?
What are the settings (flour, butter, egg) which results in the
best taste – recommendation?
What would be your approach to figure out if the cake mix is
robust against variation of the backing process?
We perform the following analysis. We use the natural
logarithm of the means.
Statistical Analysis of the Taste Rating

10-1
0,6
0,4
0,2
0,0
10-1
10-1
0,6
0,4
0,2
0,0
Flour
Mean
Butter
Egg
Corner
Center
Point Type
Main Effects Plot for LogeStd
Data Means
The parameter flour is significantThe parameter flour is significant
Main effects; Natural Logarithm Used
Stat
>DOE
>Factorial
Stat
>DOE
>Factorial
10-1 10-1
0,5
0,0
-0,5
0,5
0,0
-0,5
Flour
Butter
Egg
-1 Corner
0 Center
1 Corner
Flour Point Type
-1 Corner
0 Center
1 Corner
Butter Point Type
Interaction Plot for LogeStd
Data Means
Interactions; Natural Logarithm Used
These interactions can be
significant
These interactions can be
significant
Stat
>DOE
>Factorial
Stat
>DOE
>Factorial

Factorial Fit: LogeStd versus Flour; Butter; Egg
Estimated Effects and Coefficients for LogeStd (coded units)
Term Effect Coef SE Coef T P
Constant 0,2255 0,03601 6,26 0,025
Flour -0,6986 -0,3493 0,03819 -9,15 0,012
Butter -0,1738 -0,0869 0,03819 -2,28 0,151
Egg -0,3401 -0,1701 0,03819 -4,45 0,047
Flour*Butter -0,0328 -0,0164 0,03819 -0,43 0,710
Flour*Egg -0,1826 -0,0913 0,03819 -2,39 0,139
Butter*Egg 0,2729 0,1365 0,03819 3,57 0,070
Use column C12 for Loge
(std) as the response
Flour is significant; Eggs are on the limit;
Interaction are not significant
Flour is significant; Eggs are on the limit;
Interaction are not significant
Statistical Analysis; Natural Logarithm Used
Stat
>DOE
Stat
>DOE
What are your recommendation now?
What are the factor settings (flour, butter eggs and
backing process) in order achieve the best taste
rating?
Statistical Analysis; Natural Logarithm Used

Our king has received some information about robust designs. He wants to
know from his knights how robust is that catapult? He is interested in the
effect of three different settings on a predefined shoot distance. You have to
plan an experiment to investigate the 3 most important controllable factors!
1. Stop point (Position 2 and 4)
2. Rubber tension (Position 1 and 3)
3. Angle (155 and 170 degrees)
The noise variables are the different suppliers for the balls and the rubber
band (2 settings each).
We perform this experiment, analyze it and transfer our knowledge to the
knights and the king.
Rubber band -
fix point
Rubber band -
fix point
Stop -PositionStop -Position
Rubber band -
tension adjustment
Rubber band -
tension adjustment
1
2
345
3
4
1
2
3
4
Tension angleTension angle
6
Ball typeBall type
Rubber bandRubber band
1
2
5
6
Rubber band -
fix point
Rubber band -
fix point
Rubber band -
tension adjustment
Rubber band -
tension adjustment
1
2
345
3
4
1
2
3
4
6
Ball typeBall type
1
2
5
6
Rubber band -
fix point
Rubber band -
fix point
Rubber band -
tension adjustment
Rubber band -
tension adjustment
1
2
345
3
4
1
2
3
4
1
2
3
4
6
Ball typeBall type
1
2
5
6
Catapult Exercise
Summary
• Understanding of the principle of robust designs
within product and process development
• Understanding of the Taguchi method - application of
a DOE to investigate robustness
• Creation and analysis of experiments to investigate
factors which have the most influence on a product /
process design or variation
• Evaluation of interactions in order to make a process
robust

Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Taguchi Robust Designs

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Javier Garcia - Verdugo Sanchez - Six Sigma Training - W4 Taguchi Robust Designs