2 k factorial DOE Center Points Blocks +1 Blocks +1 F a a c t o C o r B -1 Factor A +1 -1 Week 3 Knorr-Bremse Group About this Module We know two ways to make 2 level designs We know two ways to make 2 level designs more robust and more informative. Center points to check for linearity Incorporate and evaluate additional input as a Incorporate and evaluate additional input as a block factor Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 2/27
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Javier Garcia - Verdugo Sanchez - Six Sigma Training - W32 DOE Center Points and Blocks
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2k factorial DOECenter Points
Blocks+1
Blocks+1
F aa c t o
C
o r B
-1 Factor A +1
-1 Week 3
Knorr-Bremse Group
About this Module
We know two ways to make 2 level designsWe know two ways to make 2 level designs more robust and more informative.
Center points to check for linearity
Incorporate and evaluate additional input as aIncorporate and evaluate additional input as a block factor
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 2/27
Introduction of Center Points• We always are at the risk to overlook non linear relations
within the factor settings when using DOE’s with two factor levelslevels.
• The use of center points is an effective way to easily test for linearity (curvature)for linearity (curvature).
• An example:
– A chemical engineer wants to improve the yield. Two inputs are effecting the yield: reaction time and reaction temperaturetemperature.
– The chemical engineer decides to run an 2 x 2 design and adds center points in order to proof the linearity ofand adds center points in order to proof the linearity of this model.
Inputs:– Inputs:
• Reaction temperature: 150, 155 and 160 (°C)
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 3/27
• Reaction time: 30, 35 and 40 (minutes)
The DOE with 2 Factors and 3 Center Points
StdOrder CenterPt Temp. Time Yield 11 1 150 30 39,3
R A B C Bl kExample: 2 types of catalysts as block factor Run A B C Block1 -1 -1 -1 I4 1 1 -1 I6 1 -1 1 I
Example: 2 types of catalysts as block factorRule of thumb: High order interactions seldom contribute to the model.
7 -1 1 1 I
2 1 -1 -1 II
Therefore we overlay the 3-way interaction with an additional factor (catalyst)We assign catalyst A to the low level setting of
3 -1 1 -1 II5 -1 -1 1 II8 1 1 1 II
g y gthe 3-way interaction and catalyst B to the high level. We sacrificed the 3-way interaction to save 8 runs8 runs.
Be aware: in a real experiments we like to randomize the DOE within the blocks Minitab is taking care of that
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 12/27
blocks. Minitab is taking care of that.
Blocks Confounded with Interaction
1 Type I
212 Type II
12 12+1
12C
-11 2
-1 A +1
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 13/27
-1 A +1
Evaluation of an Example
• Goal: Analyze a DOE with 4 factors and 2 blocks
• Example: A chemical engineer wants to maximize the filtration rate of a p gchemical product which is produced in a pressure vessel. For the experiment he needs 16 runs. Only 8 runs per day are possible. The complete experiment requires 2 dayscomplete experiment requires 2 days.
• Output/response: Filtration rate in l/h
• Inputs:p
– Temperature
– Pressure
– Formaldehyde concentration
– Agitation speed
– Block variable day 1 vs. day 2
• Procedure:
U th Mi it b fil Bl k filt t DOE ith 4 f t 2 bl k– Use the Minitab file: Block filter.mtw, DOE with 4 factors, 2 blocks and 16 runs
– Analyze the data
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Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 21/27
Summary
In this module we have discussed how to use:
• Center points
Bl k• Blocks
This allows us to broaden the use of 2 factor levelThis allows us to broaden the use of 2 factor level designs. As a result we are more confident regarding our statementsstatements.
Knorr-Bremse Group 02 BB W3 center points & blocks 08, D. Szemkus/H. Winkler Page 22/27
Appendix:Appendix:Evaluation of the Evaluation of the
examplep
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The Graphical Evaluation with Minitab
4,303Factor NameA C t ti
Pareto Chart of the Standardized Effects(response is Yield, Alpha = ,05)
File: 3 fact. Center points.mtw
m
AC
AB
A A C oncentrationB Rel. B/AC Temp.
Term
BC
B
C
Standardized Effect
ABC
543210
2 447
Pareto Chart of the Standardized Effects(response is Yield, Alpha = ,05)
A
2,447Factor NameA C oncentrationB Rel. B/A
Step 1: Reduce the model from the overview to get the
Term AB
gbest model. B
6543210
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Standardized Effect
The Reduced ModelF t i l Fit Yi ld C t ti R l B/AFactorial Fit: Yield versus Concentration; Rel. B/A
Estimated Effects and Coefficients for Yield (coded units)
Term Effect Coef SE Coef T PConstant 87,382 0,2573 339,65 0,000Concentration -3,500 -1,750 0,3017 -5,80 0,001Co ce t at o 3,500 , 50 0,30 5,80 0,00Rel. B/A 0,700 0,350 0,3017 1,16 0,284Concentration*Rel. B/A -0,800 -0,400 0,3017 -1,33 0,226