Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH STATISTICS FOR EXPERIMENTERS Box, Hunter and Hunter II Chapter 11 Response Surface Designs MBB_DOE_06_1 Six Sigma Master Black Belt – Chapter 11 BHH THERE HAS NEVER BEEN A SIGNAL IN THE ABSENCE OF NOISE THE GODS LOVE THE NOISE AS MUCH AS THEY LOVE THE SIGNAL LEARN TO MODEL THE SIGNAL LEARN TO MODEL THE NOISE MBB_DOE_06_2 MBB_DOE_06_1 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH THE TWO MODEL PROBLEM Every observation y has two components y = + (eta + epsilon) = the actual (true) response function being observed Examples: 0 1 x1 a straight line 0 (1 e x ) a differential equation solution 1 1 = the “noise”, “ i ” the “disturbance”, “ i ” the “error” “ ” Example: is a random event from a Normal distribution with expectation zero and variance 2 MBB_DOE_06_3 Six Sigma Master Black Belt – Chapter 11 BHH is thought to be a smooth function of the factor x1 2, the variance of the observations, the measure of noise, is unknown An experimental design: Take several equally spaced values of x1 Plan to take repeated observations at chosen settings of x1 Randomly run ALL experiments The design and observations x1 1 1 1 2 2 3 y 6.4 5.6 6.0 7.5 6.5 8.3 3 4 4 7.7 11.7 10.3 5 17.6 5 18.0 5 18.4 MBB_DOE_06_4 MBB_DOE_06_2 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_5 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_6 MBB_DOE_06_3 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_7 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_8 MBB_DOE_06_4 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH Fitted Second Order Model yˆ 8.42 3.15 x1 1.00 x12 SSq df Crude Sum of Squares y 2 1550.3000 12 Sum of Squares of Predicted Values yˆ 2 1544.5714 Residual Sum of Squares y 2 yˆ 2 5.7286 9 Lack of Fit Sum of Squares 3.4206 estimates 2 3 2 = 1.7142 = s2 if model is OK Error Sum of Squares 2.3000 Lack of fit F test: F2,7 = 7 = 0.3286 = s2 estimates 2 1.7142 0.3286 = 5.21, F2,7,0.05 = 4.79 MBB_DOE_06_9 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_10 MBB_DOE_06_5 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_11 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_12 MBB_DOE_06_6 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_13 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_14 MBB_DOE_06_7 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH Figure 11.7 (BBH, Fig 15.6, p525) MBB_DOE_06_15 Six Sigma Master Black Belt – Chapter 11 BHH 50 40 60 70 temperature a) temperature temperature b) c) Fig 11.2: a) Contours surfaces; b) 23 factorial design; c) Fitted contour surfaces. MBB_DOE_06_16 MBB_DOE_06_8 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_17 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_18 MBB_DOE_06_9 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH The model for the observations yu yu u 0 1 x1 2 x 2 u NID (0, 2 ) The fitted (1st order) model yˆ b0 b1 x1 b2 x2 ˆy 61.7 2.4 x1 3.6 x2 MBB_DOE_06_19 Six Sigma Master Black Belt – Chapter 11 BHH yˆ 6 1 .7 2 .4 x 1 3 .6 x 2 Least squares insures that y u2 yˆ u2 ( y u yˆ u ) 2 3 8 2 2 9 .3 4 3 8 2 1 8 .6 6 1 0 .6 8 Given the fitted model is appropriate s2 s2 = ( y u yˆ u ) 2 /( n p ) = 10.68/(10 - 3) = 1.54 MBB_DOE_06_20 MBB_DOE_06_10 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH ANALYSIS OF VARIANCE TABLE: THE LACK OF FIT TEST SSq q df Sum of Squares y2 38229.34 10 SSq for b0 (CF) 38068.90 1 Corrected SSq ( y y )2 160.44 9 SSq for b1 46.08 1 SSq for b2 103.68 1 Regression SSq 149.76 2 Residual SSq ( y yˆ )2 10.68 7 Lack of Fit SSq 9.00 5 1.80 = s2 ? 2 2 Error SSq 1.68 2 0.85 = s ? 2 / if model is OK F2,5 = 0.85/1.80 < 1.00, non sig, Model is OK MBB_DOE_06_21 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_22 MBB_DOE_06_11 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH Mapping the 1st order (planar) surface yˆ 61.7 61 7 22.4 4 x1 33.6 6 x2 The contour for yˆ = 60 is given by the straight line in the space of x1 and x2 60 61.7 2.4 x1 3.6 x2 Substituting yˆ = 58, 60, 62, 64, 66 gives the parallel contour lines shown in the figure MBB_DOE_06_23 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_24 MBB_DOE_06_12 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH The Path of Steepest Ascent (descent) yˆ b0 b1 x1 b2 x2 yˆ 61.7 2.4 x1 3.6 x2 Move at right angles to the contour lines For every b1 = +2.4 steps in x1 direction simultaneously take b2 = + 3.6 3 6 steps t in i the th x2 direction di ti MBB_DOE_06_25 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_26 MBB_DOE_06_13 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_27 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_28 MBB_DOE_06_14 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH SECOND ORDER MODEL 0 1 x1 2 x2 11 x12 22 x22 12 x1 x2 y NID (0, 2 ) SECOND ORDER DESIGN x1 x2 1 1 1 1 0 1 1 1 1 0 78.8 84.5 91.2 77.4 88.0 y 0 0 86.8 2 0 83.3 2 0 81.2 0 2 81.2 0 2 0 0 0 0 79.5 89.7 85.0 MBB_DOE_06_29 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_30 MBB_DOE_06_15 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_31 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_32 MBB_DOE_06_16 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_33 Six Sigma Master Black Belt – Chapter 11 BHH Mapping the 2nd order (non-planar) surface yˆ 87.38 1.38 x1 0.36 x2 2.14 x12 3.09 x22 4.88 x1 x2 The contour for yˆ = 85 is given by the curved line in the space of x1 and x2 85 0 82.38 85.0 82 38 1.38 1 38 x1 00.36 36 x2 2.14 2 14 x12 3.09 3 09 x22 4.88 4 88 x1 x2 Substituting yˆ = 80, 82.5, 85.0, 87.5 gives the curved contour lines shown in the figure MBB_DOE_06_34 MBB_DOE_06_17 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_35 Six Sigma Master Black Belt – Chapter 11 BHH Running experimental designs in blocks Suppose all the yields in Block II are 10 units less Q: What is the consequence on the response surface map? A: Almost none. The fitted model is: yˆ = 8 2 .33 8 - 1 .33 8 x 1 + 0 .33 6 x 2 - 2 .11 4 x 12 - 3 .00 9 x 22 - 4 .88 8 x 1 x 2 Only the constant term is changed In this octagon design the blocks are “Orthogonal” MBB_DOE_06_36 MBB_DOE_06_18 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_37 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_38 MBB_DOE_06_19 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_39 Six Sigma Master Black Belt – Chapter 11 BHH k = 3 factor polynomial models 1st order model 0 1 x1 2 x2 3 x3 2nd order model 0 1 x1 2 x2 3 x3 11 x12 22 x22 33 x32 12 x1 x2 13 x1 x3 23 x2 x3 MBB_DOE_06_40 MBB_DOE_06_20 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_41 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_42 MBB_DOE_06_21 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_43 Six Sigma Master Black Belt – Chapter 11 BHH The Central Composite design Sequentially assembled k = 3 0 1 x1 2 x2 3 x3 11 x12 22 x22 33 x32 12 x1 x2 13 x1 x3 23 x2 x3 x1 + + 0 0 x2 + + 0 0 x3 + + 0 0 Block I + + 0 0 + + 0 0 + + 0 0 Block II 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The central composite design can be made rotatable by choosing , the distance along the coordinate axes, equal to nc1/ 4 = 1.68. To place all points on a sphere set = 3 = 1.73. To all practical purposes either value gives orthogonal blocking. Block III MBB_DOE_06_44 MBB_DOE_06_22 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_45 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_46 MBB_DOE_06_23 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH Characteristics of a good experimental design Not too many levels Enough levels to capture the full order of the approximating polynomial model Enough points to provide for a lack of fit test Replication, or partial replication, to provide an independent p estimate of 2 Easy to block, and assemble sequentially Possess good information gathering over the experimental region (rotatability). MBB_DOE_06_47 Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_48 MBB_DOE_06_24 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_49 Six Sigma Master Black Belt – Chapter 11 BHH b) Figure 11.13b MBB_DOE_06_50 MBB_DOE_06_25 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH a) Figure 11.13a MBB_DOE_06_51 Six Sigma Master Black Belt – Chapter 11 BHH a) b) + = + = c) Figure 11.11: 23 with center points + Star = Central composite MBB_DOE_06_52 MBB_DOE_06_26 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH Figure 11.22 MBB_DOE_06_53 Six Sigma Master Black Belt – Chapter 11 BHH X1 X3 X1 X3 X2 X2 X1 X1 X3 X3 X2 X2 X1 X1 X3 X3 X2 X2 X1 X3 X2 Figure 11.19 Various three dimensional contour systems MBB_DOE_06_54 MBB_DOE_06_27 Six Sigma – Master Black Belt Advanced Design of Experiments Six Sigma Master Black Belt – Chapter 11 BHH MBB_DOE_06_55 Six Sigma Master Black Belt – Chapter 11 BHH a) b) Figure 11.14 MBB_DOE_06_56 MBB_DOE_06_28
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