2010 Stat-Ease, Inc. Dual Response Surface Methods (RSM) to Make Processes More Robust* Presented by Mark J. Anderson (
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1 Dual Response Surface Methods (RSM) to Make Processes More Robust* *Posted at Presented by Mark J. Anderson ( ) Timer by Hank Anderson July 2008 Webinar 1
2 Agenda Background k don RSM for optimizing i i processes Detail on propagation of error (POE) for finding the flats & How I applied it for driving to work The dual response approach mean & std dev Case study on dual response with POE Summary y & conclusions Suggested reading & how to get help applying these powerful statistical methods for yourself July 2008 Webinar 2
3 Response Surface Methods (RSM) When to Apply It 1. Fractional a factorials a for screening 2. High-resolution fractional or full factorial to understand interactions (add center points at this stage to test for curvature) 3. Response surface methods (RSM) to optimize. Contour maps (2D) and 3D surfaces guide you to the peak. Prediction is very hard, especially when it s about the future. - Yogi Berra July 2008 Webinar 3
4 Time is short, so hang on for a quick overview. RSM: When to Apply It (Flowchart) July 2008 Webinar 4
5 RSM: Role of Center Points If average response at the center points significantly differs from that of the outer points, add a block of axial runs to model curvature. This is a central composite design (CCD). July 2008 Webinar 5
6 Use factorial design to get close to the peak. Then RSM to climb it. RSM: When to Apply It (Topography) Region of Operability Region of Interest July 2008 Webinar 6
7 Subject Matter Knowledge (Plus Factorial Screening) Vital Few Factors (x s) Process Uncontrolled Factors (z s (zs) z s) RSM: Model Fitting Measured Response(s) (y(s)) ( Fitting Polynomial Model Response Surface July 2008 Webinar 7
8 RSM: Surfaces from Polynomials Quadratic Parabaloid Simple Cubic Sheet y = β 0 + β 1 x 12 + β 2 x 2 2 (Contains 3 rd order terms) Quadratic generally suffices, provided you get a good focus on the optimal region. July 2008 Webinar 8
9 Propagation of Error (POE) Via the tools of calculus, POE measures the variation transmitted from input factors to the response as a function of the shape of the surface. It facilitates finding the flats stable spots to locate your process, for example a high plateau of wafer yield. July 2008 Webinar 9
10 Propagation of Error Mathematical Detail Greek for the stat geeks (just kidding good stuff): 2 f f σ yˆ = σ x + σ +σ = σ i zj resid POE i xi j zj yˆ This calculation of variance (sigma-squared) squared) transmitted to the predicted response (y-hat) requires knowing the model (transfer function f(x)), typically a 2 nd order polynomial from RSM. It stems from the lack of control of the input factors (x) and known noise factors (z),* thus one must tk know (or estimate t by a swag )the standard ddeviations (sigma). The normal process variation (sigma-squared residual) is included. For convenience sake, POE is expressed as standard deviation in original units of measure hence the square root. *Consider gaining control long enough to model the impact of z factors! Goal: Minimize the propagated error (POE) July 2008 Webinar 10
11 Propagation of Error Example Calculation (p. 1/2) Find regions where variation in the control factors transmits the least variation i to the response. In this case it can be predicted d adequately by a quadratic function. ŷ =β +β x +β x yˆ = x 0.7x Goal: Minimize the slope computed by the 1 st derivative of the function. July 2008 Webinar 11
12 ŷ =β +β x +β x ŷy = x 0.7x Propagation of Error Example Calculation (p. 2/2) ˆ x resid y σ y = σ +σ x ( x ) yˆ 1 x resid σ = σ +σ Assume σ x = 1 and σ resid = 0 As the slope of the relationship between x and y decreases, the variation transmitted to y also decreases. July 2008 Webinar 12
13 POE of drive time by city slicker gets magnified by the accelerating rush of traffic at T2. POE in Real Life: Driving to Work y: Drive time T1 T x: Departure PS. This wavy curve comes from a cubic model: y = x x x July 2008 Webinar 13
14 POE minimized at flats: one high, the other low on response plot. When should commuter depart? Drive time Driving to Work: Plotting the POE POE(Dri ive time) x: Departure PS. This plot was computer-generated using ANOVA standard deviation as the floor value (see screen shot), but the POE equation could be easily calculated as detailed in referenced textbook RSM Simplified, p198. July 2008 Webinar 14
15 Dual Response Approach Analyze Mean and Standard Deviation The dual response approach suggests collecting repeated samples for each run. The average is entered as one response, and the standard deviation is entered as a second response. Goal: Use the average response to find settings that make the target product Use the standard deviation response to find settings that are robust to uncontrolled factors. This is the ONLY method that can find settings robust to unidentified sources of noise. July 2008 Webinar 15
16 Dual Response Approach How Many Samples? How Many Samples? The variability within the sample collection should represent the long-term variability of the process. The following case study by Montgomery, et al, works with only about a dozen batches, but as few as 3 samples per experimental run may be needed dto provide adequate precision. However, no matter what the sample size (n), if the study conditions are not representative of true manufacturing conditions, this method may underestimate the overall variation. Fewer samples can be used if taken over a time period that encompasses long-term variation. July 2008 Webinar 16
17 Case Study to Illustrate Dual RSM with POE: Single-Wafer Etching * Experimenters desired a more robust result for resistivity (the response output y ) as a function of three key factors (the input x s) known to affect their single-wafer etching process: A. Gas flow rate B. Temperature C. Pressure Other variables, for example radio frequency (RF) power, cannot be controlled very well. To measure the resulting variation over time, batches of wafers were collected over 11 different days from each of 17 runs in a central composite design (CCD). The process engineers hoped to hit a target resistivity of 350 ohm-cm with minimal variation. * Data from Robinson, Wulff and Montgomery, Robust Parameter Design Using Generalized Linear Mixed Models, Journal of Quality Technology, Vol. 38, No. 1, Jan. 2006, p. 70, Table 1. July 2008 Webinar 17
18 Resistivity Results (Factor levels l coded) d) Id A: Gas Flow B: Temp C: Pressure Mean Std dev July 2008 Webinar 18
19 Predictive Models for Resistivity (aka transfer functions) Mean = A B C AC B C (p<0.0001, Adj R ) Std dev = A B C BC 9.41 A B C 2 (p<0.0001, Adj R ) All models are wrong, but some are useful. G. Box Caveats (the fine print!): 1. During run #17 the result for batch #5 of 572 significantly ifi exceeded d the other 10 outcomes, which h ranged from 223 to 347. Given no special cause for this discrepant value, it was not ignored. In any case, this one value makes little difference. 2. To keep them simple, models reduced by backward regression at p of The log transformation* should be applied per standard statistical practice to standard deviations (or variances) the second of the dual responses. In this case it did not significantly improve the residuals by the Box-Cox assessment of power-law transformations, so for simplicity sake we left the results untransformed. 4. In the std dev model, the model term for C is not significant, but it is included to maintain hierarchy. * Calculation of POE for transformed responses requires some fancy footwork mathematically, involving application of the chain rule, for example. See p of RSM Simplified for details. July 2008 Webinar 19
20 Contour Plot of Resistivity Mean (Temp vs Pressure with Gas Flow at +1)* *(This factor, A, most linear set high to achieve target of y = 350) PS. Best to stay within the box of factorial settings in CCD not axials. July 2008 Webinar 20
21 Surfaces of Resistivity Mean (left) and POE* (right) (Temp vs Pressure with Gas Flow at +1) Assumes these std devs of factors (in coded scale): Gas flow (A) 0.2, Temp (B) 0.1, Pressure (C) 0.3. Wafer July 2008 Webinar 21
22 Response Surface of Resistivity Standard Deviation (Temp vs Pressure with Gas Flow at +1) Note: Generally standard deviation will be log linear, so then POE is moot. However, even when it does exhibit second-order behavior like this, it adds little or nothing to make use of the POE if it is minimized at minimal standard deviation, and nonsensical to trade it off for consistently greater variation. July 2008 Webinar 22
23 Most Desirable* Process Settings Criterion: Target mean at 350 at minimum POE with std dev (dual response) minimized * For details, see Derringer s A Balancing Act: Optimizing a Product's Properties, Quality Progress, posted at American Society for Quality (ASQ). July 2008 Webinar 23
24 3D View of Desirable Combinations of Temperature vs Pressure (gas flow at +1 level ) PS. Authors of this case study recommend coordinate (1.18, -0.80, -0.57) vs our optimal point at (1,-1,-0.5), but we do not extrapolate outside the CCD s box. July 2008 Webinar 24
25 Summary and Conclusions Response surface methods (RSM) provide statistically-validated predictive models, sometimes referred to as transfer functions, that can then be manipulated for finding optimal process configurations. Variation transmitted to responses from poorly-controlled process factors can be accounted for by the mathematical technique of propagation of error (POE), which facilitates finding the flats on the surfaces generated by RSM. The dual response approach to RSM captures the standard deviation of the output(s) as well as the average. It accounts for UNKNOWN sources of variation. Dual response plus POE provides a more useful model of overall response variation. The end-result of applying these statistical tools for design and analysis of experiments will be in-specification products that exhibit minimal variability the ultimate objective of robust design. July 2008 Webinar 25
26 Further Reading for More Detail on Methodology 1. Mark J. Anderson and Patrick J. Whitcomb, RSM Simplified Optimizing Process Using Response Surface Methods for Design of Experiments, Productivity Press, NY, NY, Mark J. Anderson and Patrick J. Whitcomb, Robust Design Reducing Transmitted Variation, Proceedings from the 50th Annual Quality Congress, 1996, pages Milwaukee: American Society of Quality. (Write-up of talk presented by MJA at the 13th SEMATECH Statistical Methods Symposium in San Antonio, TX, on April 24, 1996.) 3. Wayne A. Taylor, Comparing Three Approaches To Robust Design: Taguchi Versus Dual Response Versus Tolerance Analysis, presented at 1996 Fall Technical Conference (FTC) of the American Society of Quality (ASQ) and American Statistical Association (ASA). 4. Geoff G. Vining and Raymond H. Myers, Combining Taguchi and Response Surface Philosophies: A Dual Response Surface Approach, Journal of Quality Technology, Vol. 22, No. 1, January 1990, pp July 2008 Webinar 26
27 How to get help Search publications posted at Use Screen Tips in Stat-Ease software, view reports in annotated mode, check for context-sensitive Help (right-click) or search the main Help system. for answers from Stat-Ease s staff of statistical consultants. Call and ask for statistical help. Thanks for attending! July 2008 Webinar 27
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