Process Robustness Studies

Transcription

1 Process Robustness Studies ST 435/535 Background When factors interact, the level of one can sometimes be chosen so that another has no effect on the response. If the second factor is controllable in a test environment, but varies randomly in routine operation (a noise factor), the quality characteristic can be insulated from those variations. Genichi Taguchi developed robust parameter design to exploit this strategy. 1 / 12 Process Optimization with Designed Experiments Process Robustness Studies

2 Taguchi s approach is to construct separate designs in the controllable factors and noise factors, and to cross them. Both designs are based on orthogonal arrays, which include fractional factorial design matrices: an inner array design in the controllable factors; an outer array design in the noise factors. The crossed design uses every combination of a treatment in the controllable factors and a treatment in the noise factors. Taguchi s analysis of the resulting data differs from the conventional statistical model. 2 / 12 Process Optimization with Designed Experiments Process Robustness Studies

3 Response surface approach An alternative approach is to set up a combined design in all factors, both controllable and noise. The resulting statistical model is a prediction equation for test conditions, with all factors controllable. We assume that in routine operations the noise factors have random levels, and the statistical model provides: a mean model that predicts the mean response for given settings of the controllable factors; a variance model that shows how the variability in the noise factors propagates into the response, again for given settings of the controllable factors. 3 / 12 Process Optimization with Designed Experiments Process Robustness Studies

4 Example Suppose that x 1 and x 2 are controllable, and that z 1 is a noise factor, and the response Y satisfies a model with all main effects and all two-factor interactions, but no three-factor interaction: Y = β 0 + β 1 x 1 + β 2 x 2 + β 1,2 x 1 x 2 + γ 1 z 1 + δ 1,1 x 1 z 1 + δ 2,1 x 2 z 1 + ɛ In experiments, all three factors are controllable, and the parameters can be estimated from a factorial design. 4 / 12 Process Optimization with Designed Experiments Process Robustness Studies

5 In operations, Z 1 is random, with mean 0 and variance σ 2 Z. So and E(Y ) = β 0 + β 1 x 1 + β 2 x 2 + β 1,2 x 1 x 2 V (Y ) = σ 2 Z(γ 1 + δ 1,1 x 1 + δ 2,1 x 2 ) 2 + σ 2 ɛ. 5 / 12 Process Optimization with Designed Experiments Process Robustness Studies

6 The mean model and the variance model are used to find levels of the controllable factors that achieve some goal for the mean response (a target value, or an optimal value), while keeping variability at a satisfactory level. 6 / 12 Process Optimization with Designed Experiments Process Robustness Studies

7 Example: filtration rate The filtration rate Y in a chemical process is affected by four factors: Temperature, A; Pressure, B; Concentration, C; Stirring rate, D. Temperature is hard to control in operation: the noise factor. filtration <- expand.grid(a = c(-1, 1), B = c(-1, 1), C = c(-1, 1), D = c(-1, 1)); filtration$rate <- c(45, 71, 48, 65, 68, 60, 80, 65, 43, 100, 45, 104, 75, 86, 70, 96); library(gplots) qqnorm(aov(rate ~ A * B * C * D, filtration), label = TRUE) filtrationlm <- lm(rate ~ A * (C + D), filtration) summary(filtrationlm) 7 / 12 Process Optimization with Designed Experiments Process Robustness Studies

8 The estimated model equation is ŷ = x x 3 + ( x x 3 )z 1 so the mean model is and the variance model is Ê(Y ) = x x 3 V (Y ) = σ 2 Z( x x 3 ) 2 + σ 2 ɛ. The two levels of temperature were one standard deviation away from the mean, so σ Z = 1, and σ ɛ is estimated by the residual standard error, / 12 Process Optimization with Designed Experiments Process Robustness Studies

9 Contour plots cgrid <- dgrid <- seq(from = -1, to = 1, length = 40); m <- predict(filtrationlm, expand.grid(a = 0, C = cgrid, D = dgrid)); m <- matrix(m, length(cgrid), length(dgrid)); contour(cgrid, dgrid, m, nlevels = 5) # construct the variance model: v <- predict(filtrationlm, expand.grid(a = 1, C = cgrid, D = dgrid)); v <- matrix(v, length(cgrid), length(dgrid)); v <- (v - m)^2 + summary(filtrationlm)$sigma^2; contour(cgrid, dgrid, sqrt(v), col = "red", add = TRUE) # show a good region: image(cgrid, dgrid, ifelse (m >= 75 & sqrt(v) <= 6, TRUE, NA), col = hsv(0.33, alpha = 0.5), add = TRUE) 9 / 12 Process Optimization with Designed Experiments Process Robustness Studies

10 The goal was to achieve a filtration rate of at least 75, with low variability. The contour plot shows that this is achievable only close to the edge of the experimental domain. 10 / 12 Process Optimization with Designed Experiments Process Robustness Studies

11 Evolutionary Operation ST 435/535 The experiments involved in process improvement are usually carried out in a test facility, not the routine operational facility: off-line process improvement. The lessons learned are then implemented in routine operations: the on-line environment. Evolutionary operation consists of carrying out designed experiments in the on-line setting. 11 / 12 Process Optimization with Designed Experiments Evolutionary Operation

12 The design is typically a two-level factorial, with levels centered around current standard practice, and deviating little enough to not substantially degrade performance. Replicated center points can be added without disturbing the process. The factorial runs allow estimation of the main effects of the factors and their interactions. The center point runs allow the detection of curvature (squared terms), but not their complete characterization. 12 / 12 Process Optimization with Designed Experiments Evolutionary Operation