PracticeProblems_FractionalFactorialDesigns.pdf-10

8.51. A 16-run fractional factorial experiment in nine factors was conducted by Chrysler Motors Engineering and described in the article “Sheet Molded Compound Process Improvement,” by P.I. Hsieh and D.E. Goodwin ( Fourth Symposium on Taguchi Methods , American Supplier Institute, Dearborn, MI, 1986, pp. 13-21). The purpose was to reduce the number of defects in the finish of sheet-molded grill opening panels. The design, and the resulting number of defects, c , observed on each run, is shown in Table P8.10. This is a resolution III fraction with generators E=BD, F=BCD, G=AC, H=ACD , and J=AB. Table P8.10 Run A B C D E F G H J c c F&T’s Modification 1 - - - - + - + - + 56 7.48 7.52 2 + - - - + - - + - 17 4.12 4.18 3 - + - - - + + - - 2 1.41 1.57 4 + + - - - + - + + 4 2.00 2.12 5 - - + - + + - + + 3 1.73 1.87 6 + - + - + + + - - 4 2.00 2.12 7 - + + - - - - + - 50 7.07 7.12 8 + + + - - - + - + 2 1.41 1.57 9 - - - + - + + + + 1 1.00 1.21 10 + - - + - + - - - 0 0.00 0.50 11 - + - + + - + + - 3 1.73 1.87 12 + + - + + - - - + 12 3.46 3.54 13 - - + + - - - - + 3 1.73 1.87 14 + - + + - - + + - 4 2.00 2.12 15 - + + + + + - - - 0 0.00 0.50 16 + + + + + + + + + 0 0.00 0.50 (a) Find the defining relation and the alias relationships in this design. I = ABJ = ACG = BDE = CEF = DGH = FHJ = ABFH = ACDH = ADEJ = AEFG = BCDF = BCGJ = BEGH = CEHJ = DFGJ = ABCEH = ABDFG = ACDFJ = ADEFH = AEGHJ = BCDHJ = BCFGH = BEFGJ = CDEGJ = ABCDEG = ABCEFJ = ABDGHJ = ACFGHJ = BDEFHJ = CDEFGH = ABCDEFGHJ (b) Estimate the factor effects and use a normal probability plot to tentatively identify the important factors. Residual Normal % probability Normal plot of residuals -0.189813 -0.106313 -0.0228125 0.0606875 0.144187 1 5 10 20 30 50 70 80 90 95 99 2 2 Predicted Residuals Residuals vs. Predicted -0.189813 -0.106313 -0.0228125 0.0606875 0.144187 1.08 1.18 1.27 1.37 1.47 145

The effects are shown below in the Design Expert output. The normal probability plot of effects identifies factors A , D , F , and interactions AD , AF , BC , BG as important. Design Expert Output Term Effect SumSqr % Contribtn Model Intercept Model A -9.375 351.562 7.75573 Model B -1.875 14.0625 0.310229 Model C -3.625 52.5625 1.15957 Model D -14.375 826.562 18.2346 Error E 3.625 52.5625 1.15957 Model F -16.625 1105.56 24.3895 Model G -2.125 18.0625 0.398472 Error H 0.375 0.5625 0.0124092 Error J 0.125 0.0625 0.0013788 Model AD 11.625 540.563 11.9252 Error AE 2.125 18.0625 0.398472 Model AF 9.875 390.063 8.60507 Error AH 1.375 7.5625 0.166834 Model BC 11.375 517.563 11.4178 Model BG -12.625 637.562 14.0651 Lenth's ME 13.9775 Lenth's SME 28.3764 (c) Fit an appropriate model using the factors identified in part (b) above. The analysis of variance and corresponding model is shown below. Factors B , C , and G are included for hierarchal purposes. Design Expert Output Response: c ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 4454.13 10 445.41 28.26 0.0009 significant A 351.56 1 351.56 22.30 0.0052 B 14.06 1 14.06 0.89 0.3883 C 52.56 1 52.56 3.33 0.1274 D 826.56 1 826.56 52.44 0.0008 F 1105.56 1 1105.56 70.14 0.0004 G 18.06 1 18.06 1.15 0.3333 AD 540.56 1 540.56 34.29 0.0021 DESIGN-EXPERT Plot c A: A B: B C: C D: D E: E F: F G: G H: H J: J N o rm a l p lo t N o rm a l % p ro b a b ility E ffe c t -1 6 .6 2 -9 .5 6 -2 .5 0 4 .5 6 1 1 .6 2 1 5 1 0 2 0 3 0 5 0 7 0 8 0 9 0 9 5 9 9 A D F AD AF B C B G 146

There is a significant problem with inequality of variance. This is likely caused by the response variable being a count. A transformation may be appropriate. (e) In part (d) you should have noticed an indication that the variance of the response is not constant (considering that the response is a count, you should have expected this). The previous table also shows a transformation on c, the square root, that is a widely used variance stabilizing transformation for count data (refer to the discussion of variance stabilizing transformations in Chapter 3). Repeat parts (a) through (d) using the transformed response and comment on your results. Specifically, are the residual plots improved? Design Expert Output Term Effect SumSqr % Contribtn Model Intercept Error A -0.895 3.2041 4.2936 Model B -0.3725 0.555025 0.743752 Error C -0.6575 1.72922 2.31722 Model D -2.1625 18.7056 25.0662 Error E 0.4875 0.950625 1.27387 Model F -2.6075 27.1962 36.4439 Model G -0.385 0.5929 0.794506 Error H 0.27 0.2916 0.390754 Error J 0.06 0.0144 0.0192965 Error AD 1.145 5.2441 7.02727 Error AE 0.555 1.2321 1.65106 Error AF 0.86 2.9584 3.96436 Error AH 0.0425 0.007225 0.00968175 Error BC 0.6275 1.57502 2.11059 Model BG -1.61 10.3684 13.894 Lenth's ME 2.27978 Lenth's SME 4.62829 The analysis of the data with the square root transformation identifies only D , F , the BG interaction as being significant. The original analysis identified factor A and several two factor interactions as being significant. Residual Normal % probability Normal plot of residuals -3.8125 -1.90625 0 1.90625 3.8125 1 5 10 20 30 50 70 80 90 95 99 Predicted Residuals Residuals vs. Predicted -3.8125 -1.90625 0 1.90625 3.8125 -3.81 10.81 25.44 40.06 54.69 148

Design Expert Output Response: sqrt ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 57.42 5 11.48 6.67 0.0056 significant B 0.56 1 0.56 0.32 0.5826 D 18.71 1 18.71 10.87 0.0081 F 27.20 1 27.20 15.81 0.0026 G 0.59 1 0.59 0.34 0.5702 BG 10.37 1 10.37 6.03 0.0340 Residual 17.21 10 1.72 Cor Total 74.62 15 The Model F-value of 6.67 implies the model is significant. There is only a 0.56% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. 1.31 R-Squared 0.7694 Mean 2.32 Adj R-Squared 0.6541 C.V. 56.51 Pred R-Squared 0.4097 PRESS 44.05 Adeq Precision 8.422 Coefficient Standard 95% CI 95% CI Factor Estimate DF Error Low High VIF Intercept 2.32 1 0.33 1.59 3.05 B-B -0.19 1 0.33 -0.92 0.54 1.00 D-D -1.08 1 0.33 -1.81 -0.35 1.00 F-F -1.30 1 0.33 -2.03 -0.57 1.00 G-G -0.19 1 0.33 -0.92 0.54 1.00 BG -0.80 1 0.33 -1.54 -0.074 1.00 Final Equation in Terms of Coded Factors: sqrt = +2.32 -0.19 * B -1.08 * D -1.30 * F -0.19 * G -0.80 * B * G Final Equation in Terms of Actual Factors: sqrt = DESIGN-EXPERT Plot sqrt A: A B: B C: C D: D E: E F: F G: G H: H J: J N o rm a l p lo t N o rm a l % p ro b a b ility E ffe c t -2 .6 1 -1 .6 7 -0 .7 3 0 .2 1 1 .1 4 1 5 1 0 2 0 3 0 5 0 7 0 8 0 9 0 9 5 9 9 D F B G 149

Model BG -1.54 9.4864 14.0613 Lenth's ME 2.14001 Lenth's SME 4.34453 As with the square root transformation, factors D , F , and the BG interaction remain significant. Design Expert Output Response: F&T ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 50.00 5 10.00 5.73 0.0095 significant B 0.42 1 0.42 0.24 0.6334 D 15.92 1 15.92 9.12 0.0129 F 23.52 1 23.52 13.47 0.0043 G 0.65 1 0.65 0.37 0.5560 BG 9.49 1 9.49 5.43 0.0420 Residual 17.47 10 1.75 Cor Total 67.46 15 The Model F-value of 5.73 implies the model is significant. There is only a 0.95% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. 1.32 R-Squared 0.7411 Mean 2.51 Adj R-Squared 0.6117 C.V. 52.63 Pred R-Squared 0.3373 PRESS 44.71 Adeq Precision 7.862 Coefficient Standard 95% CI 95% CI Factor Estimate DF Error Low High VIF Intercept 2.51 1 0.33 1.78 3.25 B-B -0.16 1 0.33 -0.90 0.57 1.00 D-D -1.00 1 0.33 -1.73 -0.26 1.00 F-F -1.21 1 0.33 -1.95 -0.48 1.00 G-G -0.20 1 0.33 -0.94 0.53 1.00 BG -0.77 1 0.33 -1.51 -0.034 1.00 Final Equation in Terms of Coded Factors: F&T = +2.51 -0.16 * B -1.00 * D DESIGN-EXPERT Plot F&T A: A B: B C: C D: D E: E F: F G: G H: H J: J N o rm a l p lo t N o rm a l % p ro b a b ility E ffe c t -2 .4 2 -1 .5 3 -0 .6 3 0 .2 7 1 .1 6 1 5 1 0 2 0 3 0 5 0 7 0 8 0 9 0 9 5 9 9 D F B G 151

-1.21 * F -0.20 * G -0.77 * B * G Final Equation in Terms of Actual Factors: F&T = +2.51125 -0.16250 * B -0.99750 * D -1.21250 * F -0.20125 * G -0.77000 * B * G The following interaction plots appear as they did with the square root transformation; a slight “u” shape is observed in the residuals versus predicted plot. 8.52. An experiment is run in a semiconductor factory to investigate the effect of six factors on transistor gain. The design selected is the 2 6 2 − IV shown in Table P8.15. Table P8.15 Standard Run Order Order A B C D E F Gain 1 2 - - - - - - 1455 2 8 + - - - + - 1511 3 5 - + - - + + 1487 4 9 + + - - - + 1596 5 3 - - + - + + 1430 6 14 + - + - - + 1481 7 11 - + + - - - 1458 8 10 + + + - + - 1549 9 15 - - - + - + 1454 10 13 + - - + + + 1517 11 1 - + - + + - 1487 12 6 + + - + - - 1596 13 12 - - + + + - 1446 14 4 + - + + - - 1473 Residual Normal % probability Normal plot of residuals -2.0175 -0.99625 0.025 1.04625 2.0675 1 5 10 20 30 50 70 80 90 95 99 Predicted Residuals Residuals vs. Predicted -2.0175 -0.99625 0.025 1.04625 2.0675 -0.83 0.76 2.35 3.94 5.53 152

The Model F-value of 239.62 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. 5.88 R-Squared 0.9917 Mean 1497.75 Adj R-Squared 0.9876 C.V. 0.39 Pred R-Squared 0.9788 PRESS 885.76 Adeq Precision 44.419 Coefficient Standard 95% CI 95% CI Factor Estimate DF Error Low High VIF Intercept 1497.75 1 1.47 1494.47 1501.03 A-A 38.00 1 1.47 34.72 41.28 1.00 B-B 26.87 1 1.47 23.60 30.15 1.00 C-C -15.13 1 1.47 -18.40 -11.85 1.00 AB 13.38 1 1.47 10.10 16.65 1.00 AC -4.12 1 1.47 -7.40 -0.85 1.00 Final Equation in Terms of Coded Factors: Gain = +1497.75 +38.00 * A +26.87 * B -15.13 * C +13.38 * A * B -4.12 * A * C Final Equation in Terms of Actual Factors: Gain = +1497.75000 +38.00000 * A +26.87500 * B -15.12500 * C +13.37500 * A * B -4.12500 * A * C (c) Analyze the residuals and comment on your findings. The residual plots are acceptable. The normality and equality of variance assumptions are verified. There does not appear to be any trends or interruptions in the residuals versus run order plot. Residual Normal % probability Normal plot of residuals -7.75 -3.125 1.5 6.125 10.75 1 5 10 20 30 50 70 80 90 95 99 2 2 2 2 Predicted Residuals Residuals vs. Predicted -7.75 -3.125 1.5 6.125 10.75 1435.25 1475.25 1515.25 1555.25 1595.25 154

Run Number Residuals Residuals vs. Run -7.75 -3.125 1.5 6.125 10.75 1 4 7 10 13 16 2 2 2 2 A Residuals Residuals vs. A -7.75 -3.125 1.5 6.125 10.75 -1 0 1 2 2 2 2 2 2 B Residuals Residuals vs. B -7.75 -3.125 1.5 6.125 10.75 -1 0 1 2 2 2 2 C Residuals Residuals vs. C -7.75 -3.125 1.5 6.125 10.75 -1 0 1 2 2 D Residuals Residuals vs. D -7.75 -3.125 1.5 6.125 10.75 -1 0 1 2 2 2 2 2 2 E Residuals Residuals vs. E -7.75 -3.125 1.5 6.125 10.75 -1 0 1 155

8.53. Heat treating is often used to carbonize metal parts, such as gears. The thickness of the carbonized layer is a critical output variable from this process, and it is usually measured by performing a carbon analysis on the gear pitch (top of the gear tooth). Six factors were studied on a 2 6 2 − IV design: A = furnace temperature, B = cycle time, C = carbon concentration, D = duration of the carbonizing cycle, E = carbon concentration of the diffuse cycle, and F = duration of the diffuse cycle. The experiment is shown in Table P8.16. Table P8.16 Standard Run Order Order A B C D E F Pitch 1 5 - - - - - - 74 2 7 + - - - + - 190 3 8 - + - - + + 133 4 2 + + - - - + 127 5 10 - - + - + + 115 6 12 + - + - - + 101 7 16 - + + - - - 54 8 1 + + + - + - 144 9 6 - - - + - + 121 10 9 + - - + + + 188 11 14 - + - + + - 135 12 13 + + - + - - 170 13 11 - - + + + - 126 14 3 + - + + - - 175 15 15 - + + + - + 126 16 4 + + + + + + 193 (a) Estimate the factor effects and plot them on a normal probability plot. Select a tentative model. Design Expert Output Term Effect SumSqr % Contribtn Model Intercept Model A 50.5 10201 41.8777 Error B -1 4 0.016421 Model C -13 676 2.77515 Model D 37 5476 22.4804 DESIGN-EXPERT Plot Overlay Plot X = A: A Y = B: B Actual Factors C: C = -1.00 D: D = 0.00 E: E = 0.00 F: F = 0.00 O ve rla y P lo t A B -1 .0 0 -0 .5 0 0 .0 0 0 .5 0 1 .0 0 -1 .0 0 -0 .5 0 0 .0 0 0 .5 0 1 .0 0 G a in : 1 4 7 5 G a in : 1 5 2 5 157

Model E 34.5 4761 19.5451 Error F 4.5 81 0.332526 Error AB -4 64 0.262737 Error AC -2.5 25 0.102631 Error AD 4 64 0.262737 Error AE 1 4 0.016421 Error BD 4.5 81 0.332526 Model CD 14.5 841 3.45252 Model DE -22 1936 7.94778 Error ABD 0.5 1 0.00410526 Error ABF 6 144 0.591157 Lenth's ME 15.4235 Lenth's SME 31.3119 Factors A , C , D , E and the two factor interactions CD and DE appear to be significant. The CD and DE interactions are aliased with BF and AF interactions respectively. Because factors B and F are not significant, CD and DE were included in the model. The model can be found in the Design Expert Output below. (b) Perform appropriate statistical tests on the model. Design Expert Output Response: Pitch ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 23891.00 6 3981.83 76.57 < 0.0001 significant A 10201.00 1 10201.00 196.17 < 0.0001 C 676.00 1 676.00 13.00 0.0057 D 5476.00 1 5476.00 105.31 < 0.0001 E 4761.00 1 4761.00 91.56 < 0.0001 CD 841.00 1 841.00 16.17 0.0030 DE 1936.00 1 1936.00 37.23 0.0002 Residual 468.00 9 52.00 Cor Total 24359.00 15 The Model F-value of 76.57 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. 7.21 R-Squared 0.9808 Mean 135.75 Adj R-Squared 0.9680 C.V. 5.31 Pred R-Squared 0.9393 DESIGN-EXPERT Plot Pitch A: A B: B C: C D: D E: E F: F N o rm a l p lo t N o rm a l % p ro b a b ility E ffe c t -2 2 .0 0 -3 .8 8 1 4 .2 5 3 2 .3 8 5 0 .5 0 1 5 1 0 2 0 3 0 5 0 7 0 8 0 9 0 9 5 9 9 A C D E C D D E 158

Run Number Residuals Residuals vs. Run -13 -7.625 -2.25 3.125 8.5 1 4 7 10 13 16 2 2 A Residuals Residuals vs. A -13 -7.625 -2.25 3.125 8.5 -1 0 1 2 2 B Residuals Residuals vs. B -13 -7.625 -2.25 3.125 8.5 -1 0 1 3 3 3 C Residuals Residuals vs. C -13 -7.625 -2.25 3.125 8.5 -1 0 1 2 2 D Residuals Residuals vs. D -13 -7.625 -2.25 3.125 8.5 -1 0 1 2 2 E Residuals Residuals vs. E -13 -7.625 -2.25 3.125 8.5 -1 0 1 160