IV.4 Signal-to-Noise Ratios Background Example IV.4.1 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Background Motivation Wouldn’t it be Nice to Have a Single Performance Measure that Simultaneously Identified Factor Settings that – Optimally target the mean – Reduce variation This is the Major Motivation Underlying Taguchi’s Use of Signal-to-Noise Ratios. IV.4.2 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Background Some Popular S/N Ratios Taguchi proposed OVER 80 signal-to-noise (S/N) ratios. The following three are among his most widely applicable. Our goal is to MAXIMIZE all three. SNs = -10 log(Sy2/n) – What are the optimal values for yi? – Used when “smaller is better” SNL = -10 log(S(1/y2)/n) – What are the optimal values for yi? – Used when “larger is better” SNT = 10 log(y2/s2) – Ostensibly used when “target is better” – How does SNT measure proximity to target? IV.4.3 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Background Criticisms of Taguchi’s S/N Ratios SNs and SNL – y will almost always be a more sensitive measure of the size of effects on the mean SNT – If y and s are independent, we can look at them separately to make better decisions – y and s are frequently directly related, a situation SNT will not detect IV.4.4 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Growing an Epitaxial Layer on Silicon Wafers Figure 12 - Wafers Mounted on Susceptor Kacker, R. N. and Shoemaker, A. C. (1986). “Robust Design: A Cost-Effective Method for Improving Manufacturing Processes” AT&T Technical Journal 65, pp.311-342. IV.4.5 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Growing an Epitaxial Layer on Silicon Wafers Figure 13 - Initial and Test Settings The response variable is thickness of epitaxial layer in mm with a target of 14.5 mm. Which factors will affect – mean? – variation? Control Parameter A. Suscep tor-rotation method B. Code of Wafers C. Dep osition Temp erature D. Deposition Time E. Arsenic Flow Rate F. Hy drochloric Acid Etch Temp erature G. Hy drochloric Acid Flow Rate H. Nozzle Position Initial Setting Oscillating 1215oC Low 57% 1200oC 12% 4 Test Setting 0 Continuous 668G4 1210oC High 55% 1180oC 10% 2 Test Setting 1 Oscillating 678D4 1220oC Low 59% 1215oC 14% 6 IV.4.6 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Growing an Epitaxial Layer on Silicon Wafers Figure 14 - The Experimental Design IV.4.7 Each experimental run results in 70 observations on the response! Run Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A Cont Cont Cont Cont Cont Cont Cont Cont Osclt Osclt Osclt Osclt Osclt Osclt Osclt Osclt B 668G4 668G4 668G4 668G4 678D4 678D4 678D4 678D4 668G4 668G4 668G4 668G4 678D4 678D4 678D4 678D4 C 1210 1210 1220 1220 1210 1210 1220 1220 1210 1210 1220 1220 1210 1210 1220 1220 D High High Low Low Low Low High High Low Low High High High High Low Low E 55 59 55 59 55 59 55 59 55 59 55 59 55 59 55 59 F 1180 1215 1180 1215 1215 1180 1215 1180 1215 1180 1215 1180 1180 1215 1180 1215 G 10 14 14 10 10 14 14 10 14 10 10 14 14 10 10 14 H 2 6 6 2 6 2 2 6 2 6 6 2 6 2 2 6 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Growing an Epitaxial Layer on Silicon Wafers Figure 14 - The Experimental Design Note that the design here is “non-standard” Can you assign factors to columns A, B, C, and D in the 16-run signs table? – Hint: the original factors A, B, C and D cannot be used to generate the design Which columns would the other 4 factors be assigned to in the 16-run signs table? IV.4.8 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 - Analysis Using Only SNT Growing an Epitaxial Layer on Silicon Wafers Figure 16a - Completed Response Table y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 IV.4.9 Sum Divisor Effect A B C D SN(T) A B C E 27.84 26.37 37.25 29.6 37.25 27.8 26.65 24.05 35.45 31.5 27.8 25.77 29.35 26.31 29.75 31.49 474.23 16 29.639 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -28.45 8 -3.556 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -9.51 8 -1.189 AB AC AD -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -8.93 0.61 7.37 1.75 13.89 8 8 8 8 8 -1.116 0.0763 0.9213 0.2188 1.7363 BC 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -8.03 8 -1.004 BD 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -6.09 8 -0.761 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 - Analysis Using Only SNT Growing an Epitaxial Layer on Silicon Wafers Figure 17 - Effects Normal Probability Plot IV.4.10 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 - Analysis Using Only SNT Growing an Epitaxial Layer on Silicon Wafers Interpretation What factors favorable affect SNT? – A (susceptor rotation method) set at continuous – H (nozzle position) set at 6. IV.4.11 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Analysis Using Mean and Log(s) Growing an Epitaxial Layer on Silicon Wafers Figure 18a - Response Table for Mean y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Sum Divisor A B C D Mean A B C E 14.82 13.97 14.16 14.88 14.1 14.84 14.76 13.91 14.89 14.03 13.86 14.93 13.88 14.42 14.92 13.91 230.28 16 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -0.5 8 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 0.38 8 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -0.8 8 AB -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -0.6 8 AC 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 0.36 8 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -0.66 8 AD 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -0.02 8 BC BD 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 0.14 8 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 0.42 8 IV.4.12 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Analysis Using Mean and Log(s) Growing an Epitaxial Layer on Silicon Wafers Figure 19a - Response Table for Log(s) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 IV.4.13 Sum Divisor Effect y A B C D Log(s) A B C E -0.2213 -1 -1 -1 -0.1734 1 -1 -1 -0.7115 -1 1 -1 -0.3077 1 1 -1 -0.7154 -1 -1 1 -0.2185 1 -1 1 -0.1634 -1 1 1 -0.0595 1 1 1 -0.5995 -1 -1 -1 -0.4282 1 -1 -1 -0.2485 -1 1 -1 -0.1146 1 1 -1 -0.3253 -1 -1 1 -0.1566 1 -1 1 -0.3135 -1 1 1 -0.4313 1 1 1 -5.1882 1.4086 0.4882 0.4212 16 8 8 8 -0.32426 0.1761 0.061 0.0527 AB -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -0.047 8 -0.006 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -0.361 8 -0.045 AC 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -0.105 8 -0.013 AD 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -0.696 8 -0.087 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Analysis Using Mean and Log(s) Growing an Epitaxial Layer on Silicon Wafers Figure 20 - Effects Normal Probability Plot for Mean IV.4.14 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Analysis Using Mean and Log(s) Growing an Epitaxial Layer on Silicon Wafers Figure 21 - Effects Normal Probability Plot for Log(s) IV.4.15 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Analysis Using Mean and Log(s) Growing an Epitaxial Layer on Silicon Wafers Interpretation What factors affect the mean? – D (deposition time) set at high level increases the mean. What factor settings favorably affect variability? – A (susceptor rotation method) set at continuous. – H (nozzle position) set at 6. – D (deposition time) set at low. IV.4.16 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed. Example 6 Analysis Using Mean and Log(s) Growing an Epitaxial Layer on Silicon Wafers Interpretation Conclusions: – Set nozzle position at 6 – Use continuous susceptor rotation method – Use deposition time to adjust mean to target IV.4.17 T he Center for Reliabilty and Quality Scien ces Department of Statistics T he University of South Carolina. Copyright 199 7 D. E dwards , J. Grego and J. Lynch. All Rights Reserv ed.