Chapter 13 – Robust Design

MSE-415: B. Hawrylo

Chapter 13 – Robust Design


What is robust design/process/product?:

A robust product (process) is one that performsas intended even under non-ideal conditionssuch as process variations or a range ofoperating situations.

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What is robust design/process/product?:

For a given performance target there may bemany 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?

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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.

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P-Diagram

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 FunctionNoise Error States

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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

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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2. Formulate an objective function.An experiments performance metrics must be turned

into an objective function that relates to the desiredrobust performance.

Maximizing: Used for performance dimensionswhere larger values are better. Example?

Minimizing: Used for performance dimensionswhere smaller values are better. Example?

Target Value: Used for performance dimensionswhere values closest to a point are better. Example?

Signal-to-noise ratio: Used to measure robustness.Signal Ideal FunctionNoise Error States

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3. Develop the experimental plan.

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Step 3: DOE

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?

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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•Possibilities include

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Confounding

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

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 there are many X’s?

•What if you had this design:

•X1 – 7 levels•X2 – 5 levels•X3 – 2 levels•X4 – 2 levels

•This design would require 7x5x2x2 = 140 cases•With more variables this can get out of hand.•Fortunately DOE offers a better way.

Examples?

???

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DOE Process

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 #1All Possible

Combinations

DOE Approach #1All 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

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

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)

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

Interaction Effect

•Sometimes two factors can interact with each other.•Consider the following case

•X1 is varied over 4 levels, X2 is held constant – Y appears to be increasing as a function of X1.

1

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Interaction Effect

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

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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•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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•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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•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

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

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 DesignsOther

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: Conduct the experiment.• Vary the control and noise factors• Record the performance metrics• Compute the objective function

Step 4: Run the ExperimentStep 4: Run the Experiment

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Paper

Expt # Weight Winglet Nose Wing Trials Mean Std Dev S/N1 A1 B1 C1 D12 A1 B2 C2 D23 A1 B3 C3 D34 A2 B1 C2 D35 A2 B2 C3 D16 A2 B3 C1 D27 A3 B1 C3 D28 A3 B2 C1 D39 A3 B3 C2 D1

Airplane ExperimentAirplane Experiment

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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)

Step 5: Conduct AnalysisStep 5: Conduct Analysis

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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.)

Step 6: Select SetpointsStep 6: Select Setpoints

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Chapter 13 – Robust Design