#### Transcript DOX - University of Tennessee

IE440:PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION The 2K Factorial Design Dr. Xueping Li Dept. of Industrial & Information Engineering University of Tennessee, Knoxville 1 Design of Engineering Experiments Part 4 – Introduction to Factorials Text reference, Chapter 6 The 22 Design The 23 Design The general 2K Design The addition of center points to the 2K design 2 Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery 3 The Simplest Case: The 22 “-” and “+” denote the low and high levels of a factor, respectively Low and high are arbitrary terms Geometrically, the four runs form the corners of a square Factors can be quantitative or qualitative, although their treatment in the final model will be different 4 22 Factorial Design Involves two factors (A and B) and n replicates. We are interested in: main effect of A, main effect of B, and interaction between A and B. 5 Estimation of Factor Effects A y A y A ab a b (1) 2n 2n 21n [ab a b (1)] B yB yB ab b a (1) 2n 2n 21n [ab b a (1)] See textbook, pg. 205-206 For manual calculations The effect estimates are: A = 8.33, B = -5.00, AB = 1.67 Practical interpretation? Design-Expert analysis ab (1) a b AB 2n 2n 21n [ab (1) a b] 6 Table 6.2 (p. 208) Standard order/Yate’s order Algebraic Signs for Calculating Effects in the 22 Design 7 Contrast ::=combinations (*) signs Effects ::=2constrast/[n*2^(k)] SS ::=constrast^2/[n*2^(k)] Figure 6.1 (p. 204) Treatment combinations in the 22 design. 8 Table 6.1 (p. 208) Analysis of Variance for the Experiment in Figure 6.1 9 The Regression Model The regression model Coded variable vs. natural variable X_Coded= [Var – (Var_L+Var_H)/2 ]/(Var_H – Var_L)/2 Var = X_Coded * (Var_H – Var_L)/2 + (Var_L+Var_H)/2 10 Figure 6.2 (p. 210) Residual plots for the chemical process experiment. 11 Figure 6.3 (p. 211) Response surface plot and contour plot of yield from the chemical process experiment. 12 The 23 factorial design. Figure 6.4 (p. 211) The 23 factorial design. 13 Figure 6.5 (p. 213) Geometric presentation of contrasts corresponding to the main effects and interactions in the 23 design. 14 Table 6.3 (p. 214) Algebraic Signs for Calculating Effects in the 23 Design 15 Table 6.4 (p. 215) The Plasma Etch Experiment, Example 6.1 16 Figure 6.6 (p. 216) The 23 design for the plasma etch experiment for Example 6-1. 17 Table 6.5 (p. 217) Effect Estimate Summary for Example 6.1 18 Table 6.6 (p. 218) Analysis of Variance for the Plasma Etching Experiment 19 Figure 6.7 (p. 219) Response surface and contour plot of etch rate for Example 6-1. 20 Table 6.7a (p. 220) Design-Expert Output for Example 6.1 21 Table 6.7b (p. 221) Continued from Previous Slide 22 Table 6.7 (p. 220) Detail 1 23 Table 6.7 (p. 220) Detail 2 24 Table 6.7 (p. 220) Detail 3 25 Table 6.7 (p. 220) Detail 4 26 Table 6.7 (p. 220) Detail 5 27 Table 6.7 (p. 220) Detail 6 28 Figure 6.8 (p. 224) Ranges of etch rates for Example 6-1. 29 Table 6.8 (p. 225) Analysis Procedure for a 2k Design 30 Table 6.9 (p. 225) Analysis of Variance for a 2k Design 31 To-do-list Project 32 Figure 6.9 (p. 227) The impact of the choice of factor levels in an unreplicated design. 33 Unreplicated 2k Factorial Designs Lack of replication causes potential problems in statistical testing Replication admits an estimate of “pure error” (a better phrase is an internal estimate of error) With no replication, fitting the full model results in zero degrees of freedom for error Potential solutions to this problem Pooling high-order interactions to estimate error Normal probability plotting of effects (Daniels, 1959) Other methods…see text, pp. 234 34 Example of an Unreplicated 2k Design A 24 factorial was used to investigate the effects of four factors on the filtration rate of a resin The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate Experiment was performed in a pilot plant 35 Table 6.10 (p. 228) Pilot Plant Filtration Rate Experiment 36 Figure 6.10 (p. 228) Data from the pilot plant filtration rate experiment for Example 6-2. 37 Estimates of the Effects 38 Table 6.11 (p. 229) Contrast Constants for the 24 Design 39 Table 6.12 (p. 229) Factor Effect Estimates and Sums of Squares for the 24 Factorial in Example 6.2 40 Figure 6.11 (p. 230) Normal probability plot of the effects for the 24 factorial in Example 6-2. 41 Figure 6.12 (p. 230) Main effect and interaction plots for Example 6-2. 42 Table 6.13 (p. 231) Analysis of Variance for the Pilot Plant Filtration Rate Experiment in A, C, and D. 43 Figure 6.13 (p. 233) Normal probability plot of residuals for Example 6-2. 44 Figure 6.14 (p. 233) Contour plots of filtration rate, Example 6-2. 45 Figure 6.15 (p. 234) Half-normal plot of the factor effects from Example 6-2. 46 Figure 6.17 (p. 237) Data from the drilling experiment of Example 6-3. 47 Figure 6.18 (p. 237) Normal probability plot of effects for Example 6-3. 48 Figure 6.19 (p. 237) Normal probability plot of residuals for Example 6-3. 49 Figure 6.20 (p. 237) Plot of residuals versus predicted rate for Example 6-3. 50 Figure 6.21 (p. 238) Normal probability plot of effects for Example 6-3 following log transformation. 51 Figure 6.22 (p. 238) Normal probability plot of residuals for Example 6-3 following log transformation. 52 Figure 6.23 (p. 238) Plot of residuals versus predicted rate for Example 6-3 following log transformation. 53 Table 6.14 (p. 239) Analysis of Variance for Example 6.3 Following the Log Transformation 54 Figure 6.24 (p. 239) Data for the panel process experiment of Example 6-4. 55 Figure 6.25 (p. 239) Normal probability plot of the factor effects for the panel process experiment of Example 6-4. 56 Figure 6.26 (p. 240) Plots of residuals versus clamp time for Example 6-4. 57 Figure 6.27 (p. 240) Cube plot of temperature, clamp time, and resin flow for Example 6-4. 58 Table 6.15 (p. 241) Calculation of Dispersion Effects for Example 6.4 59 Figure 6.28 (p. 241) Normal probability plot of the dispersion effects Fi* for Example 6.4. 60 Table 6.16 (p. 242) The Oxide Thickness Experiment 61 Table 6.17 (p. 243) Effect Estimates for Example 6.5, Response Variable Is Average Oxide Thickness 62 Figure 6.29 (p. 243) Normal probability plot of the effects for the average oxide thickness response, Example 6.5. 63 Table 6.18 (p. 244) Analysis of Variance (from Design-Expert) for the Average Oxide Thickness Response, Example 6.5 64 Figure 6.30 (p. 244) Contour plots of average oxide thickness with pressure (x3) held constant. 65 Table 6.19 (p. 245) Analysis of Variance (from Design-Expert) of the Individual Wafer Oxide Thickness Response 66 Figure 6.31 (p. 246) Normal probability plot of the effects using In (s2) as the response, Example 6-5. 67 Figure 6.32 (p. 246) Contour plot of s2 (within-run variability) with pressure at the low level and gas flow at the high level. 68 Figure 6.33 (p. 246) Overlay of the average oxide thickness and s2 responses with pressure at the low level and gas flow at the high level. 69 Addition of Center Points to a 2k Designs Based on the idea of replicating some of the runs in a factorial design Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models: k k k First-order model (interaction) y 0 i xi ij xi x j i 1 k k i 1 j i k k Second-order model y 0 i xi ij xi x j ii xi2 i 1 i 1 j i i 1 70 yF yC no "curvature" The hypotheses are: k H 0 : ii 0 i 1 k H1 : ii 0 i 1 SSPure Quad nF nC ( yF yC )2 nF nC This sum of squares has a single degree of freedom 71 Central Composite Design Second-order response surface model 72 Figure 6.34 (p. 248) A 22 design with center points. 73 Figure 6.35 (p. 248) A 22 design with center points. 74 If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model 75 Table 6.20 (p. 249) Analysis of Variance for Example 6.6 76 Table 6.20 (p. 249) Detail 1 77 Table 6.20 (p. 249) Detail 2 78 Figure 6.36 (p. 250) Central composite designs. 79 Figure 6.37 (p. 251) A 22 factorial design with one qualitative factor and center points. 80 Table 6.21 (p. 252) The Circuit Experiment 81 Table 6.22 (p. 252) Regression Analysis for the Circuit Experiment Using Coded Variables 82 Table 6.23 (p. 253) Regression Analysis for the Circuit Experiment Using Engineering Units 83 Table 6.24 (p. 254) Regression Analysis for the Circuit Experiment (Interaction Term Only) 84