Full Factorial DesignsRSM EVOP
Week 3
Knorr-Bremse Group
Content
• Analysis of experiments with two or more factorAnalysis of experiments with two or more factor levels
• Use of diagnostic methods to assess the usefulness of the modelusefulness of the model
• ExamplesExamples
• Introduction to Response Surface MethodologyIntroduction to Response Surface Methodology (RSM) and to EVOP (Evolutionary Operation)
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 2/36
The Strategy of Experimentation
Collect information Fractional FactorialPlackett-Burnam
Validate factors
Analyze behavior of
Plackett BurnamFolding
2k Factorialyimportant factors
Establish a model
Center PointsBlocking
Full Factorial
Determine optimal settings
Full FactorialBox-Behnken
g
RSM EVOPRSM
Taguchi
EVOP
Knowledge and complexity define the type of experiment
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 3/36
g p y yp p
The Model for 2 Factors with 2 Levels
InteractionsMean
errorxxbxbxbbY211222110++++=
Not explainable i ti (N i )
Main effectsvariation (Noise)
The null hypothesis:
All group means are equal or similar, we cannot state a difference
Which risk are we prepared to accept?
5 10% b bilit f / i ifi l l ( 0 05 0 1)5 - 10% probability of error / significance level (α = 0,05 – 0,1)
90% power of the test (1– β ; β = 0,1)
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 4/36
Experiments with 2 Factors
22 - factorial 32 - factorial22 – factorial with center point
22 – factorial with center and star points DOE’ ith 3 f t l l lland star points DOE’s with 3 or more factor levels allow
investigation of quadratic effects.
Tenable results are obtained from 9 experimental points. Often center points
li d l ti tare realized several times to ensure correctness, estimate variation and gain degrees of freedom
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 5/36
degrees of freedom.
Design of Experiment with 3 Factors
+ + +b x x b x x b x x12 1 2 13 1 3 23 2 3y b b x b x b x= + + +0 1 1 2 2 3 3
First order model Interaction model
Second order model
+ + +b x b x b x11 12
22 22
33 32
Second order model
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 6/36
+ + +b x b x b x11 1 22 2 33 3
Design of Experiment with 3 Factors
First order design Second order design(23 factorial with center points)
g(central composite)
Box Behnken DesignBox-Behnken Design
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 7/36
Example: 2 Factors and 3 Levels
Goal: Investigate the effects of pressure and temperature on the chemical yield.chemical yield.
Output: Yield
Inputs: Temperature: 120 130 140 °CInputs: Temperature: 120, 130, 140 C
Pressure: 2, 3, 4 bar
D t
File: Full factorial 1.mtw
Data:
Temp2 3 4
pressure2 3 4
90.4 90.7 90.290.2 90.6 90.4
12090.2 90.6 90.490.1 90.5 89.990.3 90.6 90.1
130
90.5 90.8 90.490.7 90.9 90.1
140
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 8/36
The Customization of a Design in Minitab
File: Full factorial 1.mtwStat
>DOE
>F t i l>Factorial
>Define Custom Factorial Design…Temp Pressure Yield120 2 90,4120 3 90,7120 4 90,2130 2 90,1130 3 90,5130 4 89 9130 4 89,9140 2 90,5140 3 90,8140 4 90 4140 4 90,4120 2 90,2120 3 90,6120 4 90 4120 4 90,4130 2 90,3130 3 90,6130 4 90,1130 4 90,1140 2 90,7140 3 90,9140 4 90,1
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 9/36
How to Start EvaluationStat
>DOE Enter the response variable and all terms >Factorial
>Analyze Factorial Design…
p(factors and interactions).
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 10/36
The Results as an ANOVA Table
General Linear Model: Yield versus Temp; Pressure
Factor Type Levels ValuesTemp fixed 3 120; 130; 140Pressure fixed 3 2; 3; 4
Analysis of Variance for Yield, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P2 0 30111 0 30111 0 15056 8 47 0 009Temp 2 0,30111 0,30111 0,15056 8,47 0,009
Pressure 2 0,76778 0,76778 0,38389 21,59 0,000Temp*Pressure 4 0,06889 0,06889 0,01722 0,97 0,470Error 9 0,16000 0,16000 0,01778, , ,Total 17 1,29778
S = 0,133333 R-Sq = 87,67% R-Sq(adj) = 76,71%
The error term is comparably Proof the hypothesis:
Significance of temperature andsmall, that means the factors explain the variation properly.
Significance of temperature and pressure.
Interaction not significant
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 11/36
Interaction not significant
Next: Reduce the Model and Analyze Residuals
Stat
>DOE
>Factorial
>Analyze Factorial Design…
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 12/36
Result: The Reduced Model
General Linear Model: Yield versus Temp; Pressure
23 % of the variationFactor Type Levels ValuesTemp fixed 3 120; 130; 140Pressure fixed 3 2; 3; 4
is explained by temp and59 % by pressure
Analysis of Variance for Yield, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F PTemp 2 0,30111 0,30111 0,15056 8,55 0,004Pressure 2 0 76778 0 76778 0 38389 21 80 0 000Pressure 2 0,76778 0,76778 0,38389 21,80 0,000Error 13 0,22889 0,22889 0,01761Total 17 1,29778
S = 0,132691 R-Sq = 82,36% R-Sq(adj) = 76,94%
Effect plots coming next
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 13/36
Graphical Analysis
Stat
>DOE
Main effect plots or Multi-Vari Chart!
>Factorial
>Factorial Plots…
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 14/36
Graphical Presentation of the Effects
90,7Temp Pressure
Main Effects Plot for YieldData Means
Stat
>Quality Tools
90,6
90,5
Me
an
>Multi-Vari Chart…90,4
90,3
90,2
140130120
90,2
432
Multi-Vari Chart for Yield by Pressure - Temp
90,9
90,8
90,7
234
Pressure
Multi-Vari Chart for Yield by Pressure - Temp
90,6
90,5
90,4
90,3
Yie
ld
140130120
90,2
90,1
90,0
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 15/36
140130120Temp
Residual Diagnostics
Residual Plots for Yield
99
90
t
N 18AD 0,189P-Value 0,888
0,2
0,1
al
Normal Probability Plot Versus Fits
50
10
1
Per
cen
0,0
-0,1
-0,2
Res
idua
0,300,150,00-0,15-0,301
Residual90,890,690,490,290,0
Fitted Value
4 0,2
Histogram Versus Order
3
2
1
Freq
uenc
y
0,2
0,1
0,0
-0,1Res
idua
l
0,20,10,0-0,1-0,2
1
0
Residual18161412108642
-0,2
Observation Order
Normal distributed – No alarming incident
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 16/36
Another Chemical Example
• Goal: The evaluation of a 2 factor design with interaction.• Output variable: Yield
File:
Full factorial 2.mtw
• Input variable: – Temperature: 75, 80, 85 °C
C t l t t ti 5 5 6 0 6 5 %– Catalyst concentration: 5,5, 6,0, 6,5 %
• Data: TemperatureCatalyst• Data:75 80 85
76 55 5282 56 63
TemperatureCatalyst amount
5 5
Perform a complete
64 65 6587 64 6081 77 53
5,5
evaluation!
Present your results!
67 74 6383 71 6075 73 5778 86 69
6
78 86 6972 74 7085 81 6583 78 60
6,5
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 17/36
Example with 3 Factors
• Goal: Investigation of the effect of crimp, process temperature and moisture on the dye ability of nylon
Moist Crimp Temp Dye2 2 2 383 2 2 362 1 2 34moisture on the dye ability of nylon
fibers.• Output:
Dye ability (higher values are better)
Zinc2 1 2 343 1 1 282 2 2 363 1 2 35Dye ability (higher values are better)
• Inputs:– Crimp: low; high
3 1 2 351 2 2 363 1 1 273 1 3 26p ; g
– Temperature: low; medium; high– Moisture: low; medium; high
N = 3 observations per treatment
3 2 1 292 1 3 332 2 3 311 1 3 32• N = 3 observations per treatment 1 1 3 321 2 2 352 2 2 342 1 3 352 2 3 342 2 1 371 1 3 312 2 1 39
How are the optimal process settings?2 2 1 39
Any further actions you would suggest? File:
Full factorial 3 mtw
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 18/36
Full factorial 3.mtw
The Response Surface Method
Treatment of the quadratic model
Response surface graphs, wire frame and contour plot are
additional graphs to interpret resultsresults.
Th t ti f thThe computation of the quadratic model is a special
application of multipleapplication of multiple regression.
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 19/36
Example: Video Tape
• Goal: Find the settings for time and Starting DoEtemperature to optimize yield.
Time
g
• Actual settings:
Ti 75 i t
70 75 80127.5 54.3 60.3
60 3
Time
– Time = 75 minutes
– Temperature = 130°
60.3Temp 130.0 64.3
62.3132.5 64.6 68.0
File:• Starting points for the 2 factor design:
– Time: Lo = 70 minutes; Hi = 80 minutesRSM1.mtw
Time: Lo 70 minutes; Hi 80 minutes
– Temperature: Lo = 127,5°; Hi = 132,5°
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 20/36
The RSM Evaluation
Stat
>DOE
>Response Surface
>Analyze Response Surface Design…
Check if all terms are selected!selected!
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 21/36
Response Surface Regression: Yield versus Time; Temp
The RSM EvaluationResponse Surface Regression: Yield versus Time; Temp
The following terms cannot be estimated, and were removed.Temp*TempThe analysis was done using coded unitsThe analysis was done using coded units.
Estimated Regression Coefficients for Yield
Term Coef SE Coef T P
No significant interaction and no quadratic effects
Constant 62,3000 1,155 53,953 0,000Time 2,3500 1,000 2,350 0,143Temp 4,5000 1,000 4,500 0,046Time*Time -0,5000 1,528 -0,327 0,775
and no quadratic effects
Time*Temp -0,6500 1,000 -0,650 0,582
S = 2,00000 PRESS = *R-Sq = 92,93% R-Sq(pred) = *% R-Sq(adj) = 78,80%
Analysis of Variance for Yield
Source DF Seq SS Adj SS Adj MS F Pi 4 105 209 105 209 26 3021 6 58 0 136Regression 4 105,209 105,209 26,3021 6,58 0,136
Linear 2 103,090 103,090 51,5450 12,89 0,072Square 1 0,429 0,429 0,4286 0,11 0,775Interaction 1 1,690 1,690 1,6900 0,42 0,582
Residual Error 2 8 000 8 000 4 0000Residual Error 2 8,000 8,000 4,0000Pure Error 2 8,000 8,000 4,0000
Total 6 113,209
The model is represented b Y 62 3 + 2 35 time + 4 5 temp
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 22/36
The model is represented by: Y = 62,3 + 2,35 time + 4,5 temp
The Contour Plot
15080
Yield
Contour Plot of Yield vs Temp; TimeStat
>DOE
145
505560657075
86,8>Response Surface
>Contour/Surface (Wireframe) Plot…
Tem
p
140
13595
100
75808590
73,3
130
132,5
127,5
Time100959085807570
125
Following the slope starting from the area under investigation and perform further experiments on the way to the optimum.
This can be done using a graph but also using an equation.
The graph visualizes the way. This point lies far beyond of the experimental th t li t d d
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 23/36
area, the contour lines are extended
Using the Regression EquationThe equation (first-order regression model) for the yield:
Y = 62,3 + 2,35 time + 4,5 temp (in coded units)
58,0
60,0
1
Contour Plot of Yield
62,3
64,0
66,0
0mp
0
Te
10-1
-1
Time
The “easiest” way to improve the yield would be to follow the ascent which is perpendicular the contour lines. The slope we have to follow in our contour plot is
4,50/2.35 = 1,91 based on our model. That means if we make a step for time of 1 (1∆4,50/2.35 1,91 based on our model. That means if we make a step for time of 1 (1∆= 5 minutes) the step for temperature would be 1,91 (1∆ = 4,8°).
The ass mption the first order regression model is a good fit!
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 24/36
The assumption: the first-order regression model is a good fit!
Using the Regression EquationWe define now appropriate points for further experimentation. The first point is where all x’s are 0, the mean or center of the experiment. Than we choose the step size (∆) for a process variable lets say time which we callchoose the step size (∆) for a process variable, lets say time which we call ∆x1. The ∆x2 is calculated by 4,50/2,35 * ∆x1 which is 1,91. We also get now experimental points for an area where the result does not increase p panymore.
MeanTime Temperature
Time Temperature Trial YieldMeancoded coded
Time Temperature Trial Yield
Origin 0 0 75 130 6,6,7 62,3
∆ 1 1,91 5 4,8
5
Origin + 1∆ 1 1,91 80 134,8 8 73,3
Origin + 2∆ 2 3,82 85 139,6
Origin + 3∆ 3 5 73 90 144 5 9 86 8Origin + 3∆ 3 5,73 90 144,5 9 86,8
Origin + 4∆ 4 7,64 95 149,2
Origin + 5∆ 5 9,55 100 154 10 58,2
Th t i l 9 ill b f th t ti i ti t d i th t t
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 25/36
The area trial 9 will be further systematic investigated in the next step
The Next Design for the Optimized Yield
Experiments 1 - 6 have been conducted first in this example (2 levels with 2 center points) The results were confirmed by furtherlevels with 2 center points). The results were confirmed by further runs with star points and 2 center points.
Code Zeit Code Temp Yield Time Temp
-1 -1 78,8 80 140
1 1 84 5 100 140
File:
RSM3 mtw
1 -1 84,5 100 140
-1 1 91,2 80 150
1 1 77,4 100 150 RSM3.mtw0 0 89,7 90 145
0 0 86,8 90 145
1 414 0 83 3 76 145-1,414 0 83,3 76 145
1,414 0 81,2 104 145
0 -1,414 81,2 90 138
0 1,414 79,5 90 152
0 0 87 90 145
0 0 86 90 145
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 26/36
0 0 86 90 145
The Local Optimum has been Achieved Stat
>DOE
>Response Surface
>Contour/Surface (Wireframe) Plot…
Surface Plot of Yield vs Temp; Time
90Yield
Contour Plot of Yield vs Temp; Time
90
p
150,0
147,5
7072747678808284
152148
70
144
80
80 140
Yield
Temp
Tem
p
145,0
142,5
140 0
14588
886
80 14090100Time
Time10095908580
140,0
The maximum from this experiment is about 89 for the yield. Open questions: can we control the temperature in this area, does the yield
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 27/36
improvement justify the increase in energy consumption?
O ti i ti DOE 22 D i ith t i t
Optimization before Production StartOptimization DOE: 22-Design with center point
• amount NaOH amount DMS
• amount acidconcentr acid
Process Map• Addition timeHOAc
• Temperature
• amount DMS• Addition timeDMS
• Reaction time
• concentr. acid • Addition timeacid
• hydrolysis time • amount MMS
p
HADS-Formation Methylation Hydrolysis Distillation
Pareto Chart of the Effects
• amount MMS• OHMA before• DMHA before
• yield• OHMA after• DMHA after
• amount HADS
A
Pareto Chart of the Effects(response is Umsatz, Alpha = ,20)
A: NaOHB: DMS
StdOrder RunOrder CenterPt Blocks NaOH DMS Sales
B
1 1 1 1 6 60 39,01
2 2 1 1 30 60 79,84
3 3 1 1 6 72 40,15
4 4 1 1 30 72 98,35
AB
,
5 5 0 1 18 66 99,12
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 28/36
0 10 20 30 40 50
O ti i ti DOE 22 D i ith t i t
Optimization before Production Start
„robust process“
Optimization DOE: 22-Design with center point
setting20 g NaOH32 g DMS
Contour Plot of Sales
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 29/36
What does EVOP stand for ?
• A tool for process improvementp p
• A tool for capacity improvement
• Uses replicated 22 or 23 factorial DOE’s with center points• Uses replicated 22 or 23 factorial DOE s with center points
• Includes/empowers the operators
• Conducted during normal operation
• In a full scale plantp
Objective: Operate the plant to produce quality products j p p p q y pand gather information on how to improve products at the same time. The participation of many persons requires a good information flow. Small improvement steps need a big sample size.
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 30/36
EVOP - Summary
Advantages
• A simple tool for process
Disadvantages
• Limited experimental possibilities• A simple tool for process optimization for a team of responsible operators.
• Limited experimental possibilities.
• Small number of factors for every phase
• Tight experimental settings avoid process disturbance or scrap.
phase.
• Per phase often several replicates necessary
• Blocks can be used to explain other effects.
necessary.
• Needs usually several phases and th f ti
• Minor additional costs for the experimentation.
therefore more time.
• One has to be prepared for a long i t l h ith ll
• A meaningful method for continuous improvement.
experimental phase with small stepwise improvements.
With EVOP improvements will be achieved taking many small steps!
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 31/36
With EVOP improvements will be achieved taking many small steps!
The EVOP Concept Yield for several variables (Temperature, Pressure)
Example of EVOPSurface
88.088.488.8892
85.0
825 89.289.690.0904
82.5
800emp
90.480.0
77.5
Te
898825 89
89.2588.75
75.0
88.5
8988.25
88 88.4Phase1
89
89Phase2
115.0112.5110.0107.5105.0
Pressure
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 32/36
The EVOP Concept
Phase I after 5 experimental points
Cost per Ton
9.0
tio
8.5
8 0Pur
geR
at
86 918.0
7.5
Rec
ycle
:P
92 95
92
7.0
R 92 95Phase I
7.57.06.56.05.5
Reflux Ratio
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 33/36
The EVOP Concept
Phase II after 5 experimental points
Cost per Ton
9.0
tio 81808.5
8 0Pur
geR
at
82
8.0
7.5
Rec
ycle
:P 83Phase II
84
7.0
R
Phase I
7.57.06.56.05.5
Reflux Ratio
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 34/36
The EVOP Concept
Phase III after 5 experimental points
Cost per Ton
9.0
tio
8586
80
Phase III
8.5
8 0Pur
geR
at
8384
8.0
7.5
Rec
ycle
:P
Phase II
7.0
R
Phase I
7.57.06.56.05.5
Reflux Ratio
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 35/36
Summary
• Analysis of experiments with two or more factorAnalysis of experiments with two or more factor levels
• Use of diagnostic methods for the estimation of how close the model is to realityhow close the model is to reality
• ExamplesExamples
• Introduction to the Response Surface MethodIntroduction to the Response Surface Method (RSM) and to the EVOP methodology (Evolutionary Operation)(Evolutionary Operation)
Knorr-Bremse Group 04 BB W3 Full, RSM & EVOP 08, D. Szemkus/H. Winkler Page 36/36