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Chapter 13 – Robust Design
What is robust design/process/product?:
A robust product (process) is one that performs
as intended even under non-ideal conditions
such as process variations or a range of
operating situations.
!
MSE-415: B. Hawrylo
Chapter 13 – Robust Design
What is robust design/process/product?:
For a given performance target there may be
many combinations that yield desired results.
But, some combinations may be more
sensitive to variation than others.
Which combinations work the best?
Which combinations have the most ‘robust’
Design to meet customer needs?
MSE-415: B. Hawrylo
Chapter 13 – Robust Design
The robust design process has 7 primary steps:
1. Identify Signal Factor(s), Response Variable or Ideal
Function, Control Factors, Noise Factors, and Error
States (or the failure modes).
2. Formulate an objective function.
3. Develop the experimental plan.
4. Run the experiment.
5. Conduct the analysis.
6. Select and confirm factor setpoints.
7. Reflect and repeat.
MSE-415: B. Hawrylo
P-Diagram
• The P-Diagram is based on the concept of converting 100% of
input energy (input signal) into 100% of the ideal function.
• Any engineered system reaches its "ideal function" when all
of its applied energy (input) is transformed efficiently into
creating desired output energy. In reality, nothing functions
like this. Every system is less than 100% efficient in its energy
transformation. This loss goes to creating unintended
functions, or error states.
Signal = Ideal Function
Noise
Error States
MSE-415: B. Hawrylo
Chapter 13 – Robust Design
•
Identify Signal Factor(s), Response Variable or Ideal
Function, Control Factors, Noise Factors, and Error States
(or the failure modes).
1. Signal Factor (inputs) pass through the design of the product and is output into
measured Response Variable or Ideal Function.
1. The Signal Factor is transformed via the Control Factors to convert the
input to the desired Output.
2. Control Factors are typically elements such as design, materials and
processes that the engineer has 'control' over.
3. Error States are the Failure Modes or Effects of Failure as defined by an end
user when using the product.
4. Noise Factors are things that can influence the design but are not under the
control of the engineer, such as environmental factors, customer usage,
interfaces with other systems, degradation over time, piece-to-piece variation,
among others. These
1. Noise Factors, if not protected for, can make the design useless and it can
be said that the design is not robust against the expected noise factors.
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Example
• I want to determine how long it takes to boil enough
water for a of cup tea.
–
Complete the Parameter Diagram
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Chapter 13 – Robust Design
2. Formulate an objective function.
An experiments performance metrics must be turned
into an objective function that relates to the desired
robust performance.
Maximizing: Used for performance dimensions
where larger values are better. Example?
Minimizing: Used for performance dimensions
where smaller values are better. Example?
Target Value: Used for performance dimensions
where values closest to a point are better. Example?
Signal-to-noise ratio: Used to measure robustness.pg. 271
Signal = Ideal Function
Noise
Error States
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Chapter 13 – Robust Design
3. Develop the experimental plan.
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Step 3: DOE
• DOE a method for obtaining the maximum amount
of information for the least amount of data (saving
resources, money and time)
• Basic question – What is the effect on Y when
changing X?
• Simplest case: one factor, pick various levels of X
(hold other X’s constant)
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Which levels of X should be picked?
• First ask what is the range of interest?
• Statistical model valid only inside the experimental
range (Can NOT extrapolate)
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How many level of X to pick?
• What is happening between the two points?
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How many level of X to pick?
• Possibilities include
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Confounding
• Confounding is when a factorial experiment is run in blocks and the blocks
are too small to contain a complete replication of the experiment, one can
run a fraction of the replicate in each block, but the result is losing
information on some effects.
• When two factors are varied such that their individual effects can not be
determined separately, their effects are said to be confounding
• What if there are more than one independent variable? Which levels should
be picked?
• Clearly there is a trend. But is the increase in Y due to an increase in X1 or
an increase in X2 or both
• Note: In this situation, X1 and X2 are said to be confounding.
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Orthogonal Design
• A better way to set-up the experiment is to use an
orthogonal design.
• Clearly there is a trend. Increasing X1 causes Y to
increase, while increasing X2 cause Y to decrease
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What if there are many X’s?
• What if you had this design:
• X1 – 7 levels
• X2 – 5 levels
• X3 – 2 levels
• X4 – 2 levels
Examples?
???= 140 cases
• This design would require 7x5x2x2
• With more variables this can get out of hand.
• Fortunately DOE offers a better way.
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DOE Process
• Early stage of DOE

Which factors are important and what is their overall effect?
• Later stages of DOE
What is the exact relationship between each factor and its
response?
 How can we optimize combinations of factor levels to
maximize or minimize a response?

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DOE Approach #1
All Possible Combinations
• Measure effect of one particular factor by fixing
levels of remaining factors and running
experiments at various levels of factors of interest.
• Repeat entire process for each of the other factors,
one at a time.
• This allows one to measure the “simple effect” of
X’s on Y.
• Problem: too many runs needed. In our case 140
runs
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DOE approach #2
2k Factorial Design
• Choose two levels for each of k possible factors
and run experiments at each of the 2k factor-level
combinations
• Allows us to estimate the “main effect” on X’s on Y.
• Advantage: Less runs required.
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2k Factorial Design
• Two levels of factors are denoted by “-” or “low” level and
“+” or “high” level, respectively
• Example: 3 factors (23 design)
• Note: R1….R8 are values of the response associated with
the i’th combination of factor levels.
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Main Effect (ME)
• Formal definition: average change in the response
due to moving a factor from its “-” level to its “+”
level while holding all other factors fixed
• For the previous example, ME of X1 can be
calculated as follows:
• ME(X1) = (R2 – R1) + (R4 - R3) + (R6 - R5) + (R8 - R7)
4
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Interaction Effect
• Sometimes two factors can interact with each other.
• Consider the following case
1
• X1 is varied over 4 levels, X2 is held constant –
appears to be increasing as a function of X1.
Y
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Interaction Effect
• Suppose the experiment is repeated for a different
level of X2:
• Y appears to be decreasing as a function of X1 (the
exact opposite of the previous case)
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Interaction Effect
• Interactions
Effect of one factor (X1) depends on level of another factor
(X2)
 Synergistic or antagonistic
 Determine via plots or statistical test
 Higher order interactions are possible but rare

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Fractional Factorial Designs
• What if there are many factors?
Clearly this can get out of hand!
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Fractional Factorial Designs
• Is it necessary to run every single combination of
every single factor at every single level?
• Fortunately, the answer is NO?
• Fractional Factorial Design allows us to get good
estimates of main effects and interactions at a
fraction of the price or time and money!
• A certain subset of the 2k possible design points
are selected.
• But, which ones to choose?
• Theoretical statistics give the answer and
computer programs do the work.
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Fractional Factorial Designs
• How does it work?

Price of fractional factorial: certain effects confounded with
each other

Example:
– Main effect confounded with interaction:

Assumption:
– Higher order interactions are negligible main effects and
lower order interactions
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Fractional Factorial Designs
• If we can assume that higher-order interactions are
negligible, we don’t need to run every single case.
• Subject matter expert/analyst is consulted to
determine which interactions are likely to be
negligible.
• Design (run matrix) is chosen accordingly.
• DOE Strategy
Objective: screen out unimportant factors, identify significant
ones
 Another name for a fractional factorial design is “screen
design”
 Original design matrix allows us to identify significant factors –
BUT
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DOE Strategy
• What is the exact nature of the relationship
between these factors and the response?
Original design has 8 design points
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DOE Strategy
• Additional data points can be added to the original
design
• A “follow-up” study can be done to optimize or
determine sensitivity of a response.
• Other types of design exist.
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Other Designs
• Mixture/Simplex Design
Levels of X’s add up to 1 (not independent)
 Commonly used in chemistry industry
 Could be used to optimize shipfill

• Computer-Generated (Optimal) Design
Irregular experimental design (e.g., region of interest is not a
cube or a sphere due to constraints on X’s)
 Nonstandard model
 Unusual sample size requirement

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Step 4: Run the Experiment
Step 4: Conduct the experiment.
• Vary the control and noise factors
• Record the performance metrics
• Compute the objective function
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Airplane
Experiment
Paper
Expt #
1
2
3
4
5
6
7
8
9
Weight
A1
A1
A1
A2
A2
A2
A3
A3
A3
Winglet
B1
B2
B3
B1
B2
B3
B1
B2
B3
Nose
C1
C2
C3
C2
C3
C1
C3
C1
C2
Wing
D1
D2
D3
D3
D1
D2
D2
D3
D1
Trials
Mean
Std Dev
S/N
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Step 5: Conduct Analysis
Step 5: Perform analysis of means.
• Compute the mean value of the objective function
for each factor setting.
• Identify which control factors reduce the effects of
noise and which ones can be used to scale the
response. (2-Step Optimization)
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Step 6: Select Setpoints
Step 6: Select control factor setpoints.
• Choose settings to maximize or minimize objective
function.
• Consider variations carefully. (Use ANOM on variance
to understand variation explicitly.)
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