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DOE and Robust Parameter DOE and Robust Parameter Design: An Overview Design: An Overview Vijay Nair Vijay Nair University of Michigan, Ann Arbor University of Michigan, Ann Arbor [email protected] [email protected] April 4, 2006 April 4, 2006

DOE & Robust Parameter DOE & Robust Parameter

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Page 1: DOE & Robust Parameter DOE & Robust Parameter

DOE and Robust Parameter DOE and Robust Parameter Design: An OverviewDesign: An Overview

Vijay NairVijay NairUniversity of Michigan, Ann ArborUniversity of Michigan, Ann Arbor

[email protected]@umich.edu

April 4, 2006April 4, 2006

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Statistical Methods Statistical Methods for Quality and Reliabilityfor Quality and Reliability

1920s 1920s Beginnings of Modern Quality Control (Beginnings of Modern Quality Control (ShewhartShewhart))

1920s & 1930s1920s & 1930s Origins of DOE (Fisher, Yates, etc.)Origins of DOE (Fisher, Yates, etc.)

1940s (WW II)1940s (WW II) Inspection Sampling, Sequential Design, etc.Inspection Sampling, Sequential Design, etc.

1950s 1950s Work of Deming, Work of Deming, JuranJuran, Ishikawa, etc. in Japan, Ishikawa, etc. in Japan

1950s1950s Early developments in Reliability Early developments in Reliability (in Aircraft Industry (in Aircraft Industry ---- Boeing, etc.) Boeing, etc.)

1970s+80s1970s+80s Japan becomes Quality LeaderJapan becomes Quality Leader

1980s 1980s Refocus on Q&P in US and EuropeRefocus on Q&P in US and Europe

1980s1980s Quality paradigms, Taguchi, etc. in USQuality paradigms, Taguchi, etc. in US

19851985 Introduction of Introduction of Six Sigma in MotorolaSix Sigma in Motorola

1990+1990+ Continuing emphasis Continuing emphasis …… DFSS and other initiatives DFSS and other initiatives

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Industrial Applications of DOEIndustrial Applications of DOE

Factorial and fractional factorial designs (1930+)Factorial and fractional factorial designs (1930+)AgricultureAgriculture

Sequential designs (1940+) Sequential designs (1940+) DefenseDefense

Response surface designs for process optimization Response surface designs for process optimization (1950+) (1950+)

ChemicalChemical

Robust parameter design for variation reduction Robust parameter design for variation reduction (1970+) (1970+)

Manufacturing and Quality Improvement Manufacturing and Quality Improvement

Virtual (computer) experiments using computational Virtual (computer) experiments using computational models (1990+) models (1990+)

Space, Automotive, Semiconductor, Aircraft,Space, Automotive, Semiconductor, Aircraft, ……

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Design of Experiments (DOE)Design of Experiments (DOE)A key technology for optimizing product and A key technology for optimizing product and process design and for quality and reliability process design and for quality and reliability (Q&R) improvement (Q&R) improvement

Systematically investigate a system's inputSystematically investigate a system's input--output relationship to: output relationship to:

•• Improve the process (Q&R)Improve the process (Q&R)•• Identify the important design parameters Identify the important design parameters •• Optimize product or process designOptimize product or process design•• Achieve robust performanceAchieve robust performance•• Conduct accelerated stress studies for reliability Conduct accelerated stress studies for reliability

predictionprediction

•• ……

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Studying the inputStudying the input--output relationshipoutput relationshipthrough DOEthrough DOE

A

B

Y

Want to know: Effect of input parameters? Is A important?

How to manipulate A and B to optimize E(Y)? How sensitive is the optimum to changes in A and B and “noises”?

Where in the A-B region should we conduct reliability stress tests? How to extrapolate reliability results to the design

conditions?

Y = f (A, B, unknowns)

Y = f (A, B) + error

Empirical approximations to f (A,B)

Y

A, B, …

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Studying the inputStudying the input--output relationshipoutput relationshipthrough DOEthrough DOE

A

B

First-order approximation:

Y = f (A, B) + error

Empirical approximations to f (A,B)

Y

Second-order approximation

Box’s iterative philosophy

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Design of Experiments (DOE)Design of Experiments (DOE)A key technology for optimizing product and process design A key technology for optimizing product and process design and for quality and reliability (Q&R) improvement and for quality and reliability (Q&R) improvement

Systematically investigate a system's inputSystematically investigate a system's input--output output relationship to relationship to ……

Used extensively in manufacturing industriesUsed extensively in manufacturing industries

since 1980since 1980’’ss

Part of basic training programs such as SixPart of basic training programs such as Six--sigma sigma

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Six SigmaSix SigmaTypical Black Belt TrainingTypical Black Belt Training

Week 1Week 1

••Core Six SigmaCore Six Sigma••CE MatrixCE Matrix••Process CapabilityProcess Capability••Measurement SystemMeasurement System••Correlation Correlation ••Project ManagementProject Management

Week 2Week 2

••Review CapabilityReview Capability••Multivariate AnalysisMultivariate Analysis••Topics in StatisticsTopics in Statistics••Introduction to DOEIntroduction to DOE••Single Factor Single Factor ExperimentsExperiments

Week 3Week 3

••Full FactorialFull Factorial••2^k Factorials2^k Factorials••Fractional FactorialsFractional Factorials••Planning ExperimentsPlanning Experiments••EVOPEVOP••Adv. Meas. SystemsAdv. Meas. Systems

Week 4Week 4

••Advanced MultivariateAdvanced Multivariate••Multiple RegressionMultiple Regression••Response SurfaceResponse Surface••Control PlansControl Plans••Control SystemsControl Systems••Quality Function Dep.Quality Function Dep.

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If your experiment needs statistics, you ought to have done a better experiment …

Lord Rutherford

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Goals and Types of DOEGoals and Types of DOEProcess improvement Process improvement –– looking for a quick solutionlooking for a quick solution

Variable search (Variable search (ShaininShainin),, One),, One--factorfactor--atat--aa--time, time, Fractional factorial, SuperFractional factorial, Super--saturated saturated …… designsdesigns))

Screening Screening –– identify important factors from among many identify important factors from among many (Pareto principle) (Pareto principle) typically 2typically 2--level level FFDsFFDs

Product/process optimization Product/process optimization Response surface designsResponse surface designs

Achieving robustness Achieving robustness TaguchiTaguchi’’s robust parameter designss robust parameter designs

Reliability assessment and prediction Reliability assessment and prediction Accelerated stress Accelerated stress test experimentstest experiments

Virtual/Computer Experiments Virtual/Computer Experiments –– Latin hypercube, spaceLatin hypercube, space--filling, filling, …… designsdesigns

Sequential designs Sequential designs ……

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Complex Data StructureCurves, Spatial Objects, …Complex Data StructureComplex Data Structure

Curves, Spatial Objects, Curves, Spatial Objects, ……Analog signals for Analog signals for

•• 6 test conditions6 test conditions(Drive, Coast, Float, Tip(Drive, Coast, Float, Tip--In/TipIn/Tip--Out at 64 and 72 miles, Coast Out at 64 and 72 miles, Coast

Engine Off)Engine Off)•• 3 runs per test3 runs per test•• 3 Vibration signals per run3 Vibration signals per run•• 4 microphones signals per run4 microphones signals per run

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Analyzing Functional DataAnalyzing Functional DataStamping ProcessStamping Process

120 140 160 180 200 220 240-50

0

50

100

150

200

250

300

350

400

crank angle (degree)

tonnage (ton)Loose Tie Rod Worn Bearing

Excessive SnapWorn Gib

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Virtual/Computer ExperimentsVirtual/Computer Experiments

Use of computational modeling and simulation in product and Use of computational modeling and simulation in product and process design is now very commonprocess design is now very common

Design and analysis of computer experiments in very highDesign and analysis of computer experiments in very high--dimensional problems raises many interesting challenges:dimensional problems raises many interesting challenges:

•• Design strategies Design strategies Criteria? Randomness?Criteria? Randomness?

•• Goals: Understand important factors? Response surface Goals: Understand important factors? Response surface approximation? Optimization?approximation? Optimization?

•• Modeling and analysis: Use of traditional models?Modeling and analysis: Use of traditional models?

•• Model ValidationModel Validation

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TaguchiTaguchi’’s Parameter Design fors Parameter Design forAchieving Robust PerformanceAchieving Robust Performance

Product/Process

Control Factors x

Noise Factors z

Signal Factors sOutput

Y = f (x, z, s)

Target = T

Goal: Choose design factor settings to optimize performance and make system insensitive to variation in noise factors Cost-effective approach

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How?How?

Y = f( x; s; z )Y = f( x; s; z )Exploit Exploit ““interactionsinteractions”” between control factors (x) between control factors (x)

and noise factors (z) to find settings of x that and noise factors (z) to find settings of x that achieve robustness while also trying to get good achieve robustness while also trying to get good average performance.average performance.

If f(.) is known, this is a regular optimization If f(.) is known, this is a regular optimization problem. problem.

In practice, f(.) unknown, so use physical In practice, f(.) unknown, so use physical experimentation.experimentation.

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ImplementationImplementationProduct Array DesignProduct Array Design

Design for Control Factors

Control Array

Highly fractional designsMixed levels

Complex aliasingVery little focus

on CxC interactions

Noise ArraySystematically varying noise factors

Various strategies

Product Array

Can estimate all CXN interactions

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Control Factors:A – cycle time, B – mold temp, C – cavity thickness,D – holding pressure, E – injection speed, F – holding time, G – gas size

Noise Factors:M - % regrind, N - moisture content, O – ambient temp.

Injection Molding Experiment

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““Taguchi MethodsTaguchi Methods”” for Analysisfor AnalysisSNSN--Ratio for Continuous DataRatio for Continuous Data

NominalNominal--thethe--best best target value T target value T

Expected squared error loss = Expected squared error loss =

TwoTwo--stage optimization process:stage optimization process:•• Estimate SNEstimate SN--ratio and identify important ratio and identify important

““dispersiondispersion”” effects x;effects x;

•• Choose x to minimize the (estimated) SNChoose x to minimize the (estimated) SN--ratioratio•• Use Use ““adjustmentadjustment”” factors factors ““aa”” to get mean on to get mean on

target target

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AnalysisAnalysis

Half-normal plot ofLocation Effects

Half-normal plot ofDispersion Effects

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Robust Design ExamplesRobust Design ExamplesProduct Design

Water PumpResponse: Rate of water flowSignal: Input speed Control Factors: Flow pattern

Material of the pumpDesign of the impellerScroll design

Noise Factors: ContaminationsTemperature of the fluidAging

Gear SystemResponse: Output torqueSignal: Input torque Control Factors: Gear material

Number of teethType of contact

Noise Factors: Run-outType of lubricationAging

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Robust Design ApplicationsRobust Design ApplicationsProcess Design

Injection Molding ProcessResponse: Product dimensionSignal: Mold dimension Control Factors: Mold temperature

Mold materialMaterial temperatureMold pressure

Noise Factors: MoistureMold wearMaterial variability

Measurement System Design

Engine Coolant Temperature SensorResponse: Output voltageSignal: Coolant temperatureControl Factors: Various configuration of sensors

MaterialNoise Factors: Position of sensor

DegradationProduct variability

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Brief History (My version)Brief History (My version)Before 1980 Before 1980 Japan, India, Bell Labs Japan, India, Bell Labs

(~1962; (~1962; TukeyTukey; SN; SN--ratio)ratio)

TaguchiTaguchi’’s visit to Bell Labs in 1980 ***s visit to Bell Labs in 1980 ***

Activities since then:Activities since then:

AT&T, Ford, Xerox, etc AT&T, Ford, Xerox, etc ……North America, Europe, Asia North America, Europe, Asia ……ASI, Taguchi Symposia, ...ASI, Taguchi Symposia, ...Bell Labs Bell Labs MohonkMohonk Conferences (1984) Conferences (1984) QPRCQPRCNSFNSF--funded project funded project 1986 1986 visit visit Impact in Japan Impact in Japan CJQCA CJQCA Quality Progress articleQuality Progress article

Many documented examples of cost savings and Many documented examples of cost savings and process improvements process improvements

((American Supplier Institute andAmerican Supplier Institute andTaguchi Symposia Case Studies).Taguchi Symposia Case Studies).

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Early Applications at AT&TEarly Applications at AT&TWindow photolithographyWindow photolithography•• 44--fold reduction in process variancefold reduction in process variance•• 22--fold reduction in processing timefold reduction in processing time

Aluminum Etching (256K RAM)Aluminum Etching (256K RAM)•• Reduction in visual defects from 80% to 15%Reduction in visual defects from 80% to 15%

Reactive Ion EtchingReactive Ion Etching•• 50% reduction in machine utilization50% reduction in machine utilization•• $1.2M savings in machine replacement costs$1.2M savings in machine replacement costs

Film photoFilm photo--resistresist•• Reduced dropReduced drop--out rate by 50%out rate by 50%

Circuit designCircuit designWave soldering, optimum solder flux formulationWave soldering, optimum solder flux formulationRouter Bit Life ImprovementRouter Bit Life ImprovementUNIX System Response Time OptimizationUNIX System Response Time Optimization

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1986

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May or June, 1986

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@ Taguchi’s House

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Key Contributions to QualityKey Contributions to QualityIntroduce (?) robustness in process/product Introduce (?) robustness in process/product design and developmentdesign and development

Emphasis on loss Emphasis on loss vsvs specificationsspecifications

Identify sources of variation upfront:Identify sources of variation upfront:---- manufacturing, customer/environment, usage, manufacturing, customer/environment, usage, ……

Systematically introduce and study the effects of Systematically introduce and study the effects of noise factors in offnoise factors in off--line investigationsline investigations

Use this information to reduce the effect of Use this information to reduce the effect of uncontrollable noise factorsuncontrollable noise factors•• Exploit interactions between control and noise factors to Exploit interactions between control and noise factors to

achieve robustnessachieve robustness

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Contributions and Philosophy (cont.)Contributions and Philosophy (cont.)

Use DOE to study the effect of Use DOE to study the effect of ““controlcontrol”” and and ““noisenoise”” factors factors novel usenovel use

Emphasis on dispersion AND location effectsEmphasis on dispersion AND location effects

Emphasis on functionality instead of symptoms Emphasis on functionality instead of symptoms (ideal function, etc.)(ideal function, etc.)

Engineering view of DOE Engineering view of DOE –– mostly onemostly one--shot shot vsvsiterative; use of confirmation experimentsiterative; use of confirmation experiments

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Impact on IndustryImpact on IndustryWidespread recognition of the importance of Widespread recognition of the importance of robustness for variation reduction and quality robustness for variation reduction and quality improvementimprovement

Beyond parameter design Beyond parameter design –– qualitativequalitative•• EgEg., Ford Engineering ., Ford Engineering ProcessProcess development and development and

manufacturing of robust products and processes manufacturing of robust products and processes use use of systematic approach and trainingof systematic approach and training

Extensive (re)Extensive (re)--introduction, introduction, training,training, and use of DOE and use of DOE under the guise of Taguchi Methods in manufacturing under the guise of Taguchi Methods in manufacturing industries industries

ShaininShainin’’ss methods, DFSS, etc.methods, DFSS, etc.

Introduction of robustness and DOE in other Introduction of robustness and DOE in other industries (medical technology, software, industries (medical technology, software, ……) )

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““Taguchi MethodsTaguchi Methods””for Implementing Parameter Designfor Implementing Parameter Design

Emphasis on loss functions Emphasis on loss functions squared errorsquared error

Classification of problems: NominalClassification of problems: Nominal--thethe--best, smallerbest, smaller--the the better, largerbetter, larger--the better, dynamic, the better, dynamic, ……

AnalysisAnalysis•• SN ratios and twoSN ratios and two--step optimization step optimization loss functionloss function•• Various methods of analysis: accumulation, minute, Various methods of analysis: accumulation, minute,

dynamicdynamic……

Designs Designs ---- Product arrays, Product arrays, OAsOAs L_18L_18

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Issues in Experimental DesignIssues in Experimental DesignDesigns Designs ---- Product arrays, Product arrays, OAsOAs L_18L_18

Product (crossed) array Product (crossed) array vsvs Combined array Combined array •• Product array allows all c x n interactionsProduct array allows all c x n interactions•• Can get better designs or smaller run size using Can get better designs or smaller run size using

combined arrayscombined arrays•• EgEg. 4 control and 2 noise . 4 control and 2 noise 32 run PA but still only 32 run PA but still only

resolution III in control factorsresolution III in control factors•• Combined array 32 runs Combined array 32 runs Resolution VI or Resolution VI or

16 runs with Resolution IV16 runs with Resolution IV

New research on combined array designs Beyond MA designs

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TaguchiTaguchi’’s SNs SN--Ratio Analyses Ratio Analyses Biggest area of controversyBiggest area of controversy

NominalNominal--thethe--best best target value T target value T

Expected squared error loss = Expected squared error loss =

TwoTwo--stage optimization process: ***stage optimization process: ***

•• Estimate SNEstimate SN--ratio and identify important ratio and identify important ““dispersiondispersion””effects x;effects x;

•• Choose x to minimize the (estimated) SNChoose x to minimize the (estimated) SN--ratioratio•• Use Use ““adjustmentadjustment”” factors factors ““aa”” to get mean on targetto get mean on target

Similar for Similar for ““dynamicdynamic”” problemsproblems

References: In Panel Discussion (Nair, 1992), Wu References: In Panel Discussion (Nair, 1992), Wu and Hamada (2001), Techno and JQT since then.and Hamada (2001), Techno and JQT since then.

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Ensuing Discussion and ResearchEnsuing Discussion and ResearchPerMIAPerMIA•• Mathematical formulation of twoMathematical formulation of two--stage stage

optimization and development for various optimization and development for various problems and loss functions problems and loss functions (Leon et al. 1987)(Leon et al. 1987)

Generalized SNGeneralized SN--ratiosratios

Transformations Transformations (Box, 1988; Nair and (Box, 1988; Nair and PregibonPregibon, 1986), 1986)

GLM GLM ((NelderNelder and Lee, 1991)and Lee, 1991)

Dual Response Dual Response ((ViningVining and Meyers, 1990)and Meyers, 1990)

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TransformationsTransformations““VarianceVariance--stabilizingstabilizing”” transformations with no dispersion transformations with no dispersion

effects:effects:

loglog--transformation transformation

Use of BoxUse of Box--Cox transformations even with dispersion effectsCox transformations even with dispersion effectsDiagnostic: MeanDiagnostic: Mean--variance plot on logvariance plot on log--log scale:log scale:

Use slope to estimate Use slope to estimate

Advantages: Advantages: Not tied to particular loss functionNot tied to particular loss functionMore general: Does not assume gamma = 2More general: Does not assume gamma = 2DataData--analytic: estimate gamma from the dataanalytic: estimate gamma from the dataResponse surface for mean Response surface for mean ““more likelymore likely”” to be to be linear in transformed spacelinear in transformed space

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GLM GLM Joint Modeling of Location and Dispersion EffectsJoint Modeling of Location and Dispersion Effects

ComponentsComponents MeanMean DispersionDispersion

Response VariableResponse Variable YY DevianceDeviance

MeanMean

Variance FunctionVariance Function Gamma distributionGamma distribution

Link FunctionLink Function

Linear PredictorLinear Predictor

Use Extended Quasi-Likelihood criterion (Nelder and Pregibon, 1987)Iterate between mean and dispersion models

More general …Problem same as before estimating V (mu) and g (mu), …

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RemarksRemarksTaguchiTaguchi’’s SNs SN--ratios have implicit assumptions ratios have implicit assumptions and have limited validityand have limited validitySNSN--ratio and ratio and PerMIAPerMIA analyses are based on loss analyses are based on loss functionsfunctions•• Loss functions hard to specify a prioriLoss functions hard to specify a priori•• Will depend on the data metric (original Will depend on the data metric (original vsvs log, log, ……))

TwoTwo--stage optimization stage optimization Why not estimate Why not estimate mean and variance and optimize? mean and variance and optimize? Transformation and GLM based approaches more Transformation and GLM based approaches more usefulusefulJoint modeling and estimation of location and Joint modeling and estimation of location and dispersion effects intrinsically a difficult problemdispersion effects intrinsically a difficult problem

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Direct Modeling of ResponseDirect Modeling of Responseand CXN Interactionsand CXN Interactions

More generally,

Treat noise factors as fixed and absorb into structural model:Y (x) = f (control factors) + g (noise factors) + h (CxN interactions) +

-1 +1

Estimate effects of control and noisefactors and CxN interactions

Use fitted model with location and dispersioneffects to determine optimal settings for robustness and target.

Analysis more efficient treat noise factorsas fixed exploit structure of noise array

Factor A

Noise = Temp

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Other AreasOther Areas

““DynamicDynamic”” problemsproblems•• Functional responseFunctional response•• SignalSignal--response systemsresponse systems

Dynamic systemsDynamic systems

Combining robust design with controlCombining robust design with control

Probabilistic optimizationProbabilistic optimization

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Summary of Impact and ContributionsSummary of Impact and Contributions

Extensive practical impactExtensive practical impact•• Notion of robustness (qualitative)Notion of robustness (qualitative)•• Use of DOE for location and dispersionUse of DOE for location and dispersion•• Extensive use of regular DOE Extensive use of regular DOE

(more than parameter design studies)(more than parameter design studies)

ResearchResearch•• Considerable research to understand and improve on Considerable research to understand and improve on

TaguchiTaguchi’’s methods for design and analysiss methods for design and analysis•• More analysis than designMore analysis than design•• Future?Future?