Robust Design

ME 470Systems Design

Fall 2010

Why Bother?

Customers will pay for increased quality!

Customers will be loyal for increased quality!

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!

Results from Less Variation

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

less inventory!• Lower warranty costs

Taguichi Loss FunctionL

oss

TargetTarget

Traditional Approach Taguichi Definition

Why We Need to Reduce VariationC

ost

Low Variation;Minimum Cost

LSLLSL USLUSLNomNom

Cos

t

High Variation;High Cost

LSLLSL USLUSLNomNom

Cos

t

NomNom

Off target; minimum variability

USLUSLLSLLSL

Off target; barely

acceptable variability

Cos

t

NomNomLSLLSL USLUSL

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 materials– variation in manufacturing conditions– variation in manufacturing personnel– variation in the end use environment– variation in end-users– wear-out or deterioration

646362616059585756

Target USLLSL

Process Capability Analysis for Desired

% Total

% > USL

% < LSL

% Total

% > USL

% < LSL

Cpm

Ppk

PPL

PPU

Pp

StDev (Overall)

Sample N

Mean

LSL

Target

USL

0.00

0.00

0.00

0.00

0.00

0.00

2.00

2.00

2.00

2.00

2.00

0.666600

100

60

56

60

64

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

Ppk

PPL

PPU

Pp

StDev (Overall)

Sample N

Mean

LSL

Target

USL

47.40

0.00

47.40

49.00

0.00

49.00

0.32

0.02

0.02

1.67

0.84

1.57829

100

56.103

56.000

60.000

64.000

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.

Variables or parameters which– affect system performance– are uncontrollable or not economical to control

Examples include– climate– part tolerances– corrosion

Noise Factors

Classes of Noise Factors

Noise factors can be classified into:– 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

Te

mp

era

ture

(d

eg

C)

Mean=120.1

UCL=123.1

LCL=117.0

Operating Temperature

50403020100

1008

1004

1000

996

992

Observation Number

Pre

ss

ure

(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

Countermeasures for 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 system ISSUE : 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

How to describe the Engineering System?

Z1

Z2...

Zn

Y1

Y2...

Yn

X1

X2...

Xn

ControlFactors

NoiseFactors

InputsOutputs

System

The Parameter Diagram

Traditional Approach to Variation ReductionReduce 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

Classifying 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 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 this week, we used 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.

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)