Robust Design

ME 470Systems Design

Fall 2011

What are the benefits of increasing the quality of a product?

Customers will pay for increased quality!

Customers will be loyal for increased quality!

In 1980s, Ford discovered that the warranty claims on US built products were far greater than Japanese built product.

All products met the design specifications.

There was less variation in the Japanese products

There are measurable results from less variation.

• Better performance• Lower costs due to less scrap, less rework and less

inventory!• Lower warranty costs

Taguichi developed a loss function to describe the effects of variation.

Loss

TargetTarget

Traditional Approach Taguichi Definition

Why We Need to Reduce VariationC

ost Low Variation;

Minimum Cost

LSL USLNom

Cos

t

High Variation;High Cost

LSL USLNom

Cos

t

Nom

Off target; minimum variability

USLLSL

Off target; barely

acceptable variability

Cos

t

NomLSL USL

Why We Need to Shift Means

Definition of Robust DesignRobustness 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 materialsAgainst variation in manufacturing conditionsAgainst variation in manufacturing personnelAgainst variation in the end use environment

` Against variation in end-usersAgainst wear-out or deterioration

646362616059585756

Target USLLSL

Process Capability Analysis for Desired

% Total% > USL% < LSL

% Total% > USL% < LSL

Cpm

PpkPPLPPUPp

StDev (Overall)Sample NMeanLSLTargetUSL

0.000.000.00

0.000.000.00

2.00

2.002.002.002.00

0.66660010060566064

Expected PerformanceObserved Performance

Overall Capability

Process Data

If your predicted design capability looks like this, you do not have a functional performance need to apply Robust Parameter Design methods. Cost, however, may still be an issue if the input (materials or process) requirements are tight!

6462605856545250

Target USLLSL

Process Capability Analysis for Y1

% Total% > USL% < LSL

% Total% > USL% < LSL

Cpm

PpkPPLPPUPp

StDev (Overall)Sample NMeanLSL

TargetUSL

47.40 0.0047.40

49.00 0.0049.00

0.32

0.020.021.670.84

1.57829100

56.10356.000

60.00064.000

Expected PerformanceObserved Performance

Overall Capability

Process Data

If your predicted capability looks like this, you have a need to both reduce the variation and shift the mean of this characteristic - a prime candidate for the application of Robust Parameter Design methods.

Examples include climate, part tolerances, corrosion, or wear over the life of a component.

Noise Factors are variables or parameters that affect system performance and are difficult and or expensive to control.

Noise factors can be classified in many ways – customer noise, manufacturing noise, and life cycle noise can be useful classifications.

Customer usage noise Maintenance practice Geographic, climactic, cultural, and other issues Duty cycle

Manufacturing noise Processes Equipment Materials and part tolerances

Aging or life cycle noise Component wear Corrosion or chemical degradation Calibration drift

50403020100

124

123

122

121

120

119

118

117

116

Observation Number

Tem

pera

ture

(deg

C)

Mean=120.1

UCL=123.1

LCL=117.0

Operating Temperature

50403020100

1008

1004

1000

996

992

Observation Number

Pre

ssur

e (p

sia)

1

Mean=1000

UCL=1007

LCL=993.6

Pressure Variation

50403020100

82

81

80

79

78

77

76

75

Observation Number

% A

Mean=78.18

UCL=81.15

LCL=75.22

Fluid Viscosity

Operator Variation

50403020100

80

70

60

50

40

Observation Number

Dia

met

er (M

ils)

I Chart for Diameter by Operator

Mean=48.86

UCL=58.50

LCL=39.23

Operator 1 Operator 2

There are several countermeasures for dealing with noise.

Ignore them!– Will probably cause problems later on

Turn a Noise factor into a Control factor– Maintain constant temperature in the plant– Restrict operating temperature range with addition of

cooling systemISSUE : Almost always adds cost & complexity!

Compensate for effects through feedback– Adds design or process complexity

Discover and exploit opportunities to shift sensitivity– Interactions– Nonlinear relationships

The Parameter Diagram is another way to describe an Engineering System.

Z1Z2...

Zn

Y1Y2...

Yn

X1X2...

Xn

ControlFactors

NoiseFactors

InputsOutputs

System

The Parameter Diagram

The traditional approach to variation reduction is to reduce variation in X’s

What are the advantages and disadvantages of this approach?

=f( )Y

=f( )X1 X2 Xn

Y X1 X2 Xn

LSL USL

Robust Design identifies factors that cause variation in Y.

Variation in Y can be described using the mathematical model:

where Xn are Control Factors Zn are Noise Factors

ssssss nn zzzxxxyS222222 ......

2121

Factors That Have No Effects• A factor that has little or no effect on either the mean or the

variance can be termed an Economic Factor

• Economic factors should be set at a level at which costs are minimized, reliability is improved, or logistics are improved

A

2YS

Y

Main Effects Plot

Another Source of Variance Effects: Nonlinearities

ExpectedDistribution

of Y

Two Possible ControlConditions of A

Factor A has an effect on both mean and variance

Low sensitivityregion

High sensitivityregion

Summary of Variance EffectsMean Shift

Noise

A -

A +

Variance Shift

Noise

A -

A +

Mean and Variance ShiftA +

A -

Noise

Non-linearity

Robust Design Approach, 2 StepsStep 1

Reduce the variability by exploiting the active control*noise factor interactions and using a variance adjustment factor

Step 2Shift the mean to the target using a mean adjustment factor

Factorial and RSM experimental designs are used to identify the relationships required to perform these activities

Variance Shift

Noise

A -

A +

Mean Shift

Noise

B -

B +

Design Resolution• Full factorial vs. fractional factorial• In our DOE Frisbee thrower 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.

• For the screening run in lab last week, we started with a half-fractional factorial. (Say that fast 5 times!)

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.

-1, -1, -1 +1, -1, -1

+1, -1, +1

+1, +1, +1-1, +1, +1

-1, -1, +1

+1, +1, -1-1, +1, -1

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

Factors: 4 Base Design: 4, 8 Resolution: IVRuns: 16 Replicates: 2 Fraction: 1/2Blocks: 1 Center pts (total): 0

Design Generators: D = ABCAlias StructureI + ABCD

A + BCDB + ACDC + ABDD + ABCAB + CDAC + BDAD + BC

Minitab Explanation for Screening Run in Lab

Means main effects can not be distinguished from 3-ways.

Means certain 2-way interactions can not be distinguished.

A = Ball TypeB = Rubber BandsC = AngleD = Cup Position

Hubcap Example of Propagation of Errors

The example is taken from a paper presented at the Conference on Uncertainty in Engineering Design held in Gaithersburg, Maryland May10-11, 1988.

WHEELCOVER REMOVAL

WHEELCOVER RETENTION

COMPETING GOALS

OPERATIONAL GOAL

Retention Force, (N)

Retention Force, (N)

Retention Force, (N)

Retention Force, (N)