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DOE and Robust Parameter Design: An Overview

Vijay Nair

University of Michigan, Ann Arbor [email protected] April 4, 2006

Statistical Methods for Quality and Reliability

1920s 1920s & 1930s 1940s (WW II) 1950s 1950s 1970s+80s 1980s 1980s 1985 1990+ Beginnings of Modern Quality Control (Shewhart) Origins of DOE (Fisher, Yates, etc.) Inspection Sampling, Sequential Design, etc. Work of Deming, Juran, Ishikawa, etc. in Japan Early developments in Reliability (in Aircraft Industry -- Boeing, etc.) Japan becomes Quality Leader Refocus on Q&P in US and Europe Quality paradigms, Taguchi, etc. in US Introduction of Six Sigma in Motorola Continuing emphasis ... DFSS and other initiatives

Industrial Applications of DOE

Factorial and fractional factorial designs (1930+) Agriculture Sequential designs (1940+) Defense

Response surface designs for process optimization (1950+) Chemical Robust parameter design for variation reduction (1970+) Manufacturing and Quality Improvement Virtual (computer) experiments using computational models (1990+) Space, Automotive, Semiconductor, Aircraft, ...

Design of Experiments (DOE)

A key technology for optimizing product and process design and for quality and reliability (Q&R) improvement Systematically investigate a system's inputoutput relationship to:

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

·...

Studying the input-output relationship through DOE A, B, ...

Y Y

Y = f (A, B, unknowns) Y = f (A, B) + error Empirical approximations to f (A,B)

B

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?

A

Studying the input-output relationship through DOE

Y Y = f (A, B) + error Empirical approximations to f (A,B) B

First-order approximation:

A

Second-order approximation

Box's iterative philosophy

Design of Experiments (DOE)

A key technology for optimizing product and process design and for quality and reliability (Q&R) improvement Systematically investigate a system's input-output relationship to ...

Used extensively in manufacturing industries since 1980's Part of basic training programs such as Sixsigma

Six Sigma Typical Black Belt Training

Week 1

·Core Six Sigma ·CE Matrix ·Process Capability ·Measurement System ·Correlation ·Project Management

Week 2

·Review Capability ·Multivariate Analysis ·Topics in Statistics ·Introduction to DOE ·Single Factor Experiments

Week 3

·Full Factorial ·2^k Factorials ·Fractional Factorials ·Planning Experiments ·EVOP ·Adv. Meas. Systems

Week 4

·Advanced Multivariate ·Multiple Regression ·Response Surface ·Control Plans ·Control Systems ·Quality Function Dep.

If your experiment needs statistics, you ought to have done a better experiment ... Lord Rutherford

Goals and Types of DOE

Process improvement looking for a quick solution Variable search (Shainin),, One-factor-at-a-time, Fractional factorial, Super-saturated ... designs) Screening identify important factors from among many typically 2-level FFDs (Pareto principle) Product/process optimization Achieving robustness Response surface designs

Taguchi's robust parameter designs Accelerated stress

Reliability assessment and prediction test experiments

Virtual/Computer Experiments Latin hypercube, spacefilling, ... designs Sequential designs ...

Complex Data Structure Complex Data Structure Curves, Spatial Objects, ... Curves, Spatial Objects, ...

Analog signals for · 6 test conditions (Drive, Coast, Float, Tip-In/Tip-Out at 64 and 72 miles, Coast Engine Off) · 3 runs per test · 3 Vibration signals per run · 4 microphones signals per run

Analyzing Functional Data Stamping Process

tonnage (ton)

400 350 300 250 200 150 100 50 0 -50 120 140 160 180 200 220 240

Loose Tie Rod

Worn Bearing

Worn Gib Excessive Snap

crank angle (degree)

Virtual/Computer Experiments

Use of computational modeling and simulation in product and process design is now very common Design and analysis of computer experiments in very highdimensional problems raises many interesting challenges: · Design strategies Criteria? Randomness?

· Goals: Understand important factors? Response surface approximation? Optimization? · Modeling and analysis: Use of traditional models? · Model Validation

Taguchi's Parameter Design for Achieving Robust Performance

Control Factors x

Signal Factors s

Product/Process

Output Y = f (x, z, s) Target = T

Noise Factors z

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

How?

Y = f( x; s; z )

Exploit "interactions" between control factors (x) and noise factors (z) to find settings of x that achieve robustness while also trying to get good average performance. If f(.) is known, this is a regular optimization problem. In practice, f(.) unknown, so use physical experimentation.

Implementation Product Array Design

Noise Array Systematically varying noise factors Various strategies Design for Control Factors Control Array Highly fractional designs Mixed levels Complex aliasing Very little focus on CxC interactions

Product Array

Can estimate all CXN interactions

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

"Taguchi Methods" for Analysis SN-Ratio for Continuous Data Nominal-the-best target value T

Expected squared error loss = Two-stage optimization process:

· Estimate SN-ratio and identify important "dispersion" effects x; · Choose x to minimize the (estimated) SN-ratio · Use "adjustment" factors "a" to get mean on target

Analysis

Half-normal plot of Location Effects

Half-normal plot of Dispersion Effects

Robust Design Examples

Product Design

Water Pump

Response: Rate of water flow Signal: Input speed Control Factors: Flow pattern Material of the pump Design of the impeller Scroll design Noise Factors: Contaminations Temperature of the fluid Aging

Gear System

Response: Output torque Signal: Input torque Control Factors: Gear material Number of teeth Type of contact Noise Factors: Run-out Type of lubrication Aging

Robust Design Applications

Process Design

Injection Molding Process Response: Product dimension Signal: Mold dimension Control Factors: Mold temperature Mold material Material temperature Mold pressure Noise Factors: Moisture Mold wear Material variability

Measurement System Design

Engine Coolant Temperature Sensor Response: Output voltage Signal: Coolant temperature Control Factors: Various configuration of sensors Material Noise Factors: Position of sensor Degradation Product variability

Brief History (My version)

Before 1980 Japan, India, Bell Labs (~1962; Tukey; SN-ratio) Taguchi's visit to Bell Labs in 1980 *** Activities since then:

AT&T, Ford, Xerox, etc ... North America, Europe, Asia ... ASI, Taguchi Symposia, ... Mohonk Conferences (1984) QPRC Bell Labs NSF-funded project 1986 visit Impact in Japan CJQCA Quality Progress article Many documented examples of cost savings and process improvements (American Supplier Institute and Taguchi Symposia Case Studies).

Early Applications at AT&T

Window photolithography

· 4-fold reduction in process variance · 2-fold reduction in processing time

Aluminum Etching (256K RAM)

· Reduction in visual defects from 80% to 15%

Reactive Ion Etching

· 50% reduction in machine utilization · $1.2M savings in machine replacement costs

Film photo-resist

· Reduced drop-out rate by 50%

Circuit design Wave soldering, optimum solder flux formulation Router Bit Life Improvement UNIX System Response Time Optimization

1986

May or June, 1986

@ Taguchi's House

Key Contributions to Quality

Introduce (?) robustness in process/product design and development Emphasis on loss vs specifications Identify sources of variation upfront:

-- manufacturing, customer/environment, usage, ...

Systematically introduce and study the effects of noise factors in off-line investigations Use this information to reduce the effect of uncontrollable noise factors

· Exploit interactions between control and noise factors to achieve robustness

Contributions and Philosophy (cont.)

Use DOE to study the effect of "control" and "noise" factors novel use Emphasis on dispersion AND location effects Emphasis on functionality instead of symptoms (ideal function, etc.) Engineering view of DOE mostly one-shot vs iterative; use of confirmation experiments

Impact on Industry

Widespread recognition of the importance of robustness for variation reduction and quality improvement Beyond parameter design qualitative

development and · Eg., Ford Engineering Process manufacturing of robust products and processes use of systematic approach and training

Extensive (re)-introduction, training, and use of DOE under the guise of Taguchi Methods in manufacturing industries Shainin's methods, DFSS, etc. Introduction of robustness and DOE in other industries (medical technology, software, ...)

"Taguchi Methods" for Implementing Parameter Design

Emphasis on loss functions squared error

Classification of problems: Nominal-the-best, smaller-the better, larger-the better, dynamic, ... Analysis · SN ratios and two-step optimization loss function · Various methods of analysis: accumulation, minute, dynamic... Designs -- Product arrays, OAs L_18

Issues in Experimental Design

Designs -- Product arrays, OAs L_18 Product (crossed) array vs Combined array

· Product array allows all c x n interactions · Can get better designs or smaller run size using combined arrays 32 run PA but still only · Eg. 4 control and 2 noise resolution III in control factors Resolution VI or · Combined array 32 runs 16 runs with Resolution IV

New research on combined array designs Beyond MA designs

Taguchi's SN-Ratio Analyses Biggest area of controversy

Nominal-the-best target value T

Expected squared error loss = Two-stage optimization process: ***

· Estimate SN-ratio and identify important "dispersion" effects x; · Choose x to minimize the (estimated) SN-ratio · Use "adjustment" factors "a" to get mean on target

Similar for "dynamic" problems References: In Panel Discussion (Nair, 1992), Wu and Hamada (2001), Techno and JQT since then.

Ensuing Discussion and Research

PerMIA

· Mathematical formulation of two-stage optimization and development for various problems and loss functions

(Leon et al. 1987)

Generalized SN-ratios Transformations

(Box, 1988; Nair and Pregibon, 1986)

GLM

(Nelder and Lee, 1991) (Vining and Meyers, 1990)

Dual Response

Transformations

"Variance-stabilizing" transformations with no dispersion effects: log-transformation Use of Box-Cox transformations even with dispersion effects Diagnostic: Mean-variance plot on log-log scale:

Use slope to estimate Advantages: Not tied to particular loss function More general: Does not assume gamma = 2 Data-analytic: estimate gamma from the data Response surface for mean "more likely" to be linear in transformed space

GLM Joint Modeling of Location and Dispersion Effects

Components Response Variable Mean Variance Function Link Function Linear Predictor Gamma distribution Mean Y Dispersion Deviance

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), ...

Remarks

Taguchi's SN-ratios have implicit assumptions and have limited validity SN-ratio and PerMIA analyses are based on loss functions

· Loss functions hard to specify a priori · Will depend on the data metric (original vs log, ...)

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

Direct Modeling of Response and 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) +

Estimate effects of control and noise factors and CxN interactions Use fitted model with location and dispersion effects to determine optimal settings for robustness and target. Analysis more efficient treat noise factors as fixed exploit structure of noise array -1 +1 Factor A

Noise = Temp

Other Areas

"Dynamic" problems

· Functional response · Signal-response systems

Dynamic systems Combining robust design with control Probabilistic optimization

Summary of Impact and Contributions

Extensive practical impact

· · · Notion of robustness (qualitative) Use of DOE for location and dispersion Extensive use of regular DOE (more than parameter design studies)

Research

· Considerable research to understand and improve on Taguchi's methods for design and analysis · More analysis than design · Future?

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