This document discusses the principles and methods of robust design. It defines robustness as a product or process being minimally affected by sources of variation. Taguchi methods are presented for designing products and processes that are robust, including parameter design to optimize the signal-to-noise ratio. An example applies Taguchi methods to the design of a Nerf missile to maximize firing distance while minimizing the impact of noise factors like user skill level. The results of the experiment are analyzed using Minitab to identify the optimal settings for the design parameters to achieve the most robust performance.

1. Robust Design
ME 470
Systems Design
Fall 2005

2. Why Bother?
Customers will pay for
increased quality!
Customers will be loyal
for increased quality!

3. Taguchi Case Study
In 1980s, Ford outsourced the construction
of a subassembly to several of its own
plants and to a Japanese manufacturer.
Both US and Japan plants produced parts
that conformed to specification (zero
defects)
Warranty claims on US built products was
far greater!!!
The difference? Variation
Japanese product was far more consistent!

4. Results from Less Variation
Better performance
Lower costs due to less scrap, less rework and
less inventory!
Lower warranty costs

5. Taguichi Loss Function
Target Target
Traditional Approach Taguichi Definition

6. Taguichi’s Loss Function
Quality Level = total loss All products suffer some
incurred by society due to loss
the failure of the product The smaller the loss the
to deliver the expected better
performance + harmful Emphasis shifts from being
side effects (including within a range to
operating cost) achieving the target
value

7. L( y) = k ( y − m) 2
(12.7)
L(y) is the quality lost (often measured in$)
y is value of the quality characteristic
m is the target value for y
k is the quality loss coefficient

8. Variations of the quadratic loss
function
Nominal is best (Figure 12.6)
Smaller the better (Figure 12.6)
Larger the better (Figure 12.6)
An asymmetric relationship (we won’t worry
about this)

9. Noise
Output variability Input variability
– Variational noise is the – Tolerances are design
short term, unit to unit factor variability
variation due to – Outer noise represents
manufacturing processes variations in disturbance
– Inner noise is the long- factors such as
term change in product temperature, humidity,
characteristics over time dust, etc

10. Signal to noise ratio
Attempts to control the variation with respect
to both the mean and the variation about the
mean.
Appropriate S/N ratios are given in equation
12.12, 12.13, and 12.14 for nominal is best,
smaller is best, and larger is best. Be careful
that you use the correct equation.
The HIGHER the S/N ratio, the better!

11. Definition of Robust Design
Robustness is defined as a condition in which the product
or process will be minimally affected by sources of
variation.
A product can be robust:
Against variation in raw materials
Against variation in manufacturing conditions
Against variation in manufacturing personnel
Against variation in the end use environment
` Against variation in end-users
Against wear-out or deterioration

12. Back to M&M’s®
The making of M&M's® Milk Chocolates begins with milk chocolate centers,
which are formed in a machine and then quot;tumbledquot; in order to obtain a
smooth, rounded center.
What follows is a process known as quot;panningquot;. Panning involves coating the
chocolates by rotating them in a coating material in a revolving pan.
Panning can be done using syrups and other materials such as chocolate,
fats etc. The principle, briefly, is to coat the center with a layer of
materials, which on evaporation leaves an even layer or shell of dry
substance. The chocolate centers are color coated by rotating them in a
revolving pan, while a sugar and corn syrup mixture is added. This
process is repeated several times until M&M's® have a thin, smooth shell
with the desired thickness.
Then, the machine specially designed for the purpose gently imprints an 'm'
on the surface of the fragile, crispy, colorful shell without cracking the
shell.

13. What are Sources of Noise?
Output variability Input variability

14. Parameter Design
Taguchi recommends parameter design to get
the best S/N ratio. If parameter design is not
sufficient, then tolerance design may be used.
Look for two types of design factors
– control factors affect the S/N ratio, but not the mean
– signal factors affect primarily the mean
Taguchi creates a design parameter matrix and
a noise matrix

15. Parameter Design
The book has an excellent example in Section
12.6.1
If tolerance design is necessary, typically
ANOVA is used to determine the relative
contribution of each control parameter
Some STATISTICIANS hate Taguchi!!
But Taguchi is used by many companies!!

16. Design Resolution
Full factorial vs. fractional factorial
In our DOE experiment, we used a full factorial.
This can become costly as the number of
variables or levels increases.
As a result, statisticians use fractional factorials.
As you might suspect, you do not get as much
information from a fractional factorial.

17. Fractional Factorials
A Fractional Factorial Design is a factorial design
in which all possible treatment combinations of
the factors are NOT run. The runs are just a
FRACTION of the full factorial matrix. The
resulting design matrix will not be able to
estimate some of the effects, often the
interaction effects. Minitab and your statistics
textbook will tell you the form necessary for
fractional factorials.

18. Design Resolution
Resolution V (Best)
– Main effects are confounded with 4-way interactions
– 2-way interactions are confounded with 3-way interactions
Resolution IV
– Main effects are confounded with 3-way interactions
– 2-way interactions are confounded with other 2-way
interactions
Resolution III (many Taguchi arrays)
– Main effects are confounded with 2-way interactions
– 2-way interactions may be confounded with other 2-ways

19. Test Schedules Available in Text
Figure 12. 9 (a) L9 experimental design for 4
control factors, each at three levels.
Figure 12.9 (b) L4 experimental design for 3
factors each at 2 levels.
Example 12.6 Three output temperatures,
each at three levels. Use Figure 12.9 (a) and
omit a column.

20. Procedure for Taguchi
Determine Design Parameters (Inner array)
Determine Noise factors (Outer array)
Select the appropriate test matrix.
Run the experiment
Analyze the results

21. Taguchi Example of Robust Design
Nerf Missilestorm
Design Parameters
– fin arrangement,
– cavity size,
– lubrication
Noise Factors
– skill of user
Performance Characteristic
Example modified from – firing distance
Product Design; Techniques in Reverse
Engineering and New Product
Development

22. Form the inner array or the
design parameter matrix
Design parameters (2 ) High (1) Low
d1 is the fin arrangement Angled fins Straight
d2 is the lubrication Graphite powder None
d3 is the cavity size 0 cm 3 cm
Determine the Required Number of Experiments:
One degree of freedom is associated with the overall mean. Next we
add the degrees of freedom associated with each design parameter: (#
of levels of design parameters)*(# of design parameters).
For our case =3*(2-1) or 3. Therefore, we must run at least 1+3 tests.
Test schedules may be found in different texts. We will only use the
ones in our book or Minitab.

23. Test Matrix from 12.9 b
Experiment 1 2 3
#
1 1 1 1
2 1 2 2
3 2 1 2
4 2 2 1

24. Design Parameter Matrix
Trial d1 d2 d3 Randomized Order
1 1 1 1 3
2 1 2 2 2
3 2 1 2 4
4 2 2 1 1

25. Outer array, Noise Factor matrix
Since there is only one Noise Factor in this experiment,
we have a column matrix.
n1
1 Unskilled Operator
2 Skilled Operator
We will have to conduct 8 experiments. Each experiment
will have to be conducted at both levels of the noise factor.

26. Results of Tests: Max Distance (m)
Noise Noise
Trial low (1) high(2)
1 9.30 8.31
2 9.83 10.11
3 8.26 6.99
4 8.23 9.19

27. Calculate S/N ratio for each row
Because we want to maximize distance, we use Equation 12.14.
1 1
S / N = −10 * log( ∑ 2 )
n y i
So for Trial 1,
1 1 1
S / N = −10 * log( ( 2 + 2
))
2 9.3 8.31

28. Signal to Noise Ratios for Maximum
Distance Tests
Trial # S/N
1 18.85
2 19.97
3 17.55
4 18.76

29. Analysis of Results
Calculate the average S/N for each factor at each of the three
test levels.
Average Average S/N
of 1 & 2
Level d1 d2 d3
1 19.4 18.2 18.8
2 18.2 19.4 18.8
Average of Trials
___ & ___

30. Firgure 1: Comparison of S/N Ratio's for each
Design Parameter
19.4
19.4
19.0
S/N Ratio
18.8 d1
18.8 d2
d3
18.2 18.2
18.0
0 1 2 3
Level
Use Largest S/N –
Pick d1 at Level 1

31. Let’s Do It in Minitab
>Stat>DOE>Taguchi>Create Taguchi Design
Select
Display
Available
Designs

32. Select the
L4 Design

33. Select
factors

34. Enter
Factor
Names

35. Minitab generates
runs that must be
completed
You must enter
noise data directly

37. >Stat>DOE>Taguchi>Analyze Taguchi Design
Select
analysis
Double Click on
Unskilled Operator
and Skilled Operator

39. Always check Options!!!! Minitab will
assume a default value which may cause you
to miss 10 points on the final.

40. Main Effects Plot (data means) for SN ratios
Fin Arrangement Lubrication
19.6
19.2
18.8
Mean of SN ratios
18.4
18.0
1 2 1 2
Cavity Size
19.6
19.2
18.8
18.4
18.0
1 2
Signal-to-noise: Larger is better

41. Response Table for Signal to Noise Ratios
Larger is better
Cavity
Level Fin Arrangement Lubrication Size
1 19.41 18.20 18.81
2 18.16 19.37 18.76
Delta 1.25 1.16 0.04
Rank 1 2 3

42. Homework: Due Monday, October 30
Individual Assignment
12.9 from Dieter by hand
Modified 12.11 must be done using Minitab