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DOE & Robust Parameter DOE & Robust Parameter

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- 1. DOE and Robust Parameter Design: An Overview Vijay Nair University of Michigan, Ann Arbor vnn@umich.edu April 4, 2006
- 2. Statistical Methods for Quality and Reliability 1920s Beginnings of Modern Quality Control (Shewhart) 1920s & 1930s Origins of DOE (Fisher, Yates, etc.) 1940s (WW II) Inspection Sampling, Sequential Design, etc. 1950s Work of Deming, Juran, Ishikawa, etc. in Japan 1950s Early developments in Reliability (in Aircraft Industry -- Boeing, etc.) 1970s+80s Japan becomes Quality Leader 1980s Refocus on Q&P in US and Europe 1980s Quality paradigms, Taguchi, etc. in US 1985 Introduction of Six Sigma in Motorola 1990+ Continuing emphasis … DFSS and other initiatives
- 3. 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, …
- 4. 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: • 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 •…
- 5. Studying the input-output relationship through DOE A, B, … Y Y Y = f (A, B, unknowns) B Y = f (A, B) + error Empirical approximations to f (A,B) A 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?
- 6. Studying the input-output relationship through DOE Y Y = f (A, B) + error Empirical approximations B to f (A,B) First-order approximation: A Second-order approximation Box’s iterative philosophy
- 7. 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 Six- sigma
- 8. Six Sigma Typical Black Belt Training Week 1 Week 2 •Core Six Sigma •Review Capability •CE Matrix •Multivariate Analysis •Process Capability •Topics in Statistics •Measurement System •Introduction to DOE •Correlation •Single Factor Experiments •Project Management Week 3 Week 4 •Full Factorial •Advanced Multivariate •2^k Factorials •Multiple Regression •Fractional Factorials •Response Surface •Planning Experiments •Control Plans •EVOP •Control Systems •Adv. Meas. Systems •Quality Function Dep.
- 9. If your experiment needs statistics, you ought to have done a better experiment … Lord Rutherford
- 10. 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 (Pareto principle) typically 2-level FFDs Product/process optimization Response surface designs Achieving robustness Taguchi’s robust parameter designs Reliability assessment and prediction Accelerated stress test experiments Virtual/Computer Experiments – Latin hypercube, space- filling, … designs Sequential designs …
- 11. 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
- 12. Analyzing Functional Data Stamping Process tonnage (ton) 400 Loose Tie Rod Worn Bearing 350 300 250 200 Worn Gib 150 Excessive Snap 100 50 0 -50 120 140 160 180 200 220 240 crank angle (degree)
- 13. 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 high- dimensional 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
- 14. Taguchi’s Parameter Design for Achieving Robust Performance Control Factors x Output Product/Process Signal Factors s 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
- 15. 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.
- 16. Implementation Product Array Design Noise Array Systematically varying noise factors Various strategies Design for Control Factors Control Array Product Array Highly fractional designs Can estimate all Mixed levels CXN interactions Complex aliasing Very little focus on CxC interactions
- 17. 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
- 18. “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
- 19. Analysis Half-normal plot of Half-normal plot of Location Effects Dispersion Effects
- 20. 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
- 21. 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
- 22. 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, ... Bell Labs Mohonk Conferences (1984) QPRC 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).
- 23. 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
- 24. 1986
- 25. May or June, 1986
- 26. @ Taguchi’s House
- 27. 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
- 28. 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
- 29. Impact on Industry Widespread recognition of the importance of robustness for variation reduction and quality improvement Beyond parameter design – qualitative • Eg., Ford Engineering Process development and 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, …)
- 30. “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
- 31. 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 • Eg. 4 control and 2 noise 32 run PA but still only resolution III in control factors • Combined array 32 runs Resolution VI or 16 runs with Resolution IV New research on combined array designs Beyond MA designs
- 32. 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.
- 33. 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) Dual Response (Vining and Meyers, 1990)
- 34. 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
- 35. GLM Joint Modeling of Location and Dispersion Effects Components Mean Dispersion Response Variable Y Deviance Mean Variance Function Gamma distribution Link Function Linear 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), …
- 36. 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
- 37. 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 Noise = Temp 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
- 38. Other Areas “Dynamic” problems • Functional response • Signal-response systems Dynamic systems Combining robust design with control Probabilistic optimization
- 39. 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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