Download - S3 - Process product optimization design experiments response surface methodolgy - Session 3/4
![Page 1: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/1.jpg)
Process/product optimization using design of experiments and response surface methodology
Mikko Mäkelä
Sveriges landbruksuniversitetSwedish University of Agricultural Sciences
Department of Forest Biomaterials and TechnologyDivision of Biomass Technology and ChemistryUmeå, Sweden
![Page 2: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/2.jpg)
Contents
Practical course, arranged in 4 individual sessions:
Session 1 – Introduction, factorial design, first order models
Session 2 – Matlab exercise: factorial design
Session 3 – Central composite designs, second order models, ANOVA,
blocking, qualitative factors
Session 4 – Matlab exercise: practical optimization example on given data
![Page 3: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/3.jpg)
Session 1
Introduction
Why experimental design
Factorial design
Design matrix
Model equation = coefficients
Residual
Response contour
![Page 4: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/4.jpg)
Session 2
Factorial design
Research problem
Design matrix
Model equation = coefficients
Degrees of freedom
Predicted response
Residual
ANOVA
R2
Response contour
![Page 5: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/5.jpg)
Session 3
Central composite designs
Design variance
Common designs
Second order models
Stationary points
ANOVA
Blocking
Confounding
Qualitative factors
![Page 6: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/6.jpg)
Central composite designs
f(x) f(x)
x1 x2 x1 x2x3
First order f(x) Second order f(x)
![Page 7: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/7.jpg)
Central composite designs
Second order models through
Center-points
Axial points
αnc
![Page 8: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/8.jpg)
Central composite designs
Center-points (nc)
Pure error (lack of fit)
Curvature
Axial points (α)
Quadratic terms
Spherical designα > 1
Cuboidal designα = 1
![Page 9: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/9.jpg)
Central composite designs
Design characteristics nc and α
Pure error (lack of fit)
Estimated error distribution
Area of operability
Control over factor levels
![Page 10: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/10.jpg)
Central composite designs
Practical design optimality
Model parameters (βi)
Prediction ( ) quality
Prediction ( ) quality emphasized
Design rotatability
r
[0, 0]
SPV = f(r)
Scaled prediction variance (SPV):
SPVNVar x
σ
![Page 11: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/11.jpg)
Central composite designs
CCD, k 2, 2, 5
CCD, k 2, 2, 1
Scaled prediction variance
![Page 12: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/12.jpg)
Central composite designs
Common designs
Central composite α > 1
![Page 13: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/13.jpg)
Central composite designs
Common designs
Central composite α = 1
![Page 14: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/14.jpg)
Central composite designs
Common designs
Box-Behnken
![Page 15: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/15.jpg)
Second order models
First order models
Main effects
Main effects + interactions
Second order models
Main effects + interactions + quadratic terms
⋯
![Page 16: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/16.jpg)
Second order models
N:o xi xj xij xii xjj
1 -1 -1 1 12 1 -1 -1 13 -1 1 -1 14 1 1 1 15 -α 0 0 06 α 0 0 07 0 -α 0 α2
8 0 α 0 α2
9 0 0 0 010 0 0 0 011 0 0 0 0
Design matrix, k = 2
Factorial
Axial
Center-points
![Page 17: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/17.jpg)
Research problem
A central composite design was
performed for a tire tread compound
Two factors x1 and x2
Axial distance α = 1.633
N:o of center-point nc = 4
Measured response, y
Tire abrasion index
Factor Factor levelsx1 -1.633 -1 0 1 1.633x2 -1.633 -1 0 1 1.633
Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 275.
![Page 18: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/18.jpg)
Research problemN:o x1 x2 x12 x11 x22 y
1 -1 -1 1 1 1 2702 1 -1 -1 1 1 2703 -1 1 -1 1 1 3104 1 1 1 1 1 2405 -1.633 0 0 2.667 0 5506 1.633 0 0 2.667 0 2607 0 -1.633 0 0 2.667 5208 0 1.633 0 0 2.667 3809 0 0 0 0 0 52010 0 0 0 0 0 29011 0 0 0 0 0 58012 0 0 0 0 0 590
Factorial
Axial
Center-points
![Page 19: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/19.jpg)
Research problem
Unrefined coefficients
Contour
![Page 20: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/20.jpg)
Second order models
Second order models can include stationary points:
Saddle point Maximum/minimum
![Page 21: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/21.jpg)
Second order models
Stationary point character can be described
Fitted second order model (k = 2)
Derivation 0 results in
2 0
2 0
![Page 22: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/22.jpg)
Second order models
For analysing a stationary point
′ where
⋯ , ⋮ and
/2 ⋯ /2⋯ /2⋱ ⋮
sym.
→ location and character
![Page 23: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/23.jpg)
Second order models
Stationary point location
From the previous example
0.5 0.2
485.8
![Page 24: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/24.jpg)
Second order models
Stationary point character
/2 ⋯ /2⋯ /2⋱ ⋮
sym.
Eigenvalues
, , ⋯ , all < 0 → Maximum
, , ⋯ , all > 0 → Minimum
, , ⋯ , mixed in sign → Saddle point..
![Page 25: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/25.jpg)
Coefficients
Response dependent of a coefficient
H0: ⋯ 0
H1: 0 for at least one j
Lack of fit
Corrected cp residuals vs. others
→ Sufficiently fitted model?
ANOVA
![Page 26: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/26.jpg)
ANOVA based on the F test
Tests if two sample populations
have equal variances (H0)
Ratio of variances and respective dfs
Distribution for every combination of dfs
One- or two-tailed
Alternative hypothesis (H1)
upper one-tailed (reject H0 if F F ∝, df , df )
ANOVA
![Page 27: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/27.jpg)
ANOVA
Parameter df Sum of squares (SS)
Meansquare (MS) F-value p-value
Total corrected n-1 SStot MStot
Regression k SSmod MSmod MSmod/MSres
<0.05>0.05
Residual n-p SSres MSres
Lack of fitn-p-
(nc-1) SSlof MSlof MSlof/MSpe
<0.05>0.05
Pure error nc-1 SSpe MSpe
p = k + 1
MS = SS / df
![Page 28: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/28.jpg)
Research problem
An extraction process (x1,x2,x3) was studied using a cuboidal central
composite design (α = 1, nc = 3) for maximizing yield
Statistically significant coefficients x1, x2, x3 and x12
Responses (in order): 56.6, 58.5, 48.9, 55.2, 61.8, 63.3, 61.5, 64, 61.3,
65.5, 64.6, 65.9, 63.6, 65.0, 62.9, 63.8, 63.5
Present a full ANOVA table
Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 266.
![Page 29: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/29.jpg)
Research problem
Sum of squares for pure error
SS of center-points corrected for the (center-point) mean
Parameter df Sum of squares (SS)
Meansquare (MS) F-value p-value
Total corrected
Regression
Residual
Lack of fit
Pure error
![Page 30: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/30.jpg)
ANOVA
Response transformations or modification of model terms might alleviate
lack of fit
![Page 31: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/31.jpg)
Blocking
Blocking/confounding can be used to separate nuisance effects Different batches of raw materials
Varying conditions on different days
Blocking
Replicated designs arranged in different blocks
Confounding
A single design divided into different blocks
→ 2k design in 2p blocks where p < k
In a 23 design with 2 blocks, confound nuisance to x123
![Page 32: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/32.jpg)
N:o x1 x2 x3 x123 y1 - - - - 902 + - - + 643 - + - + 814 + + - - 635 - - + + 776 + - + - 617 - + + - 888 + + + + 53
Blocking
E.g. 2 blocks based on the x123 interaction (randomized within blocks)
Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 126.
![Page 33: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/33.jpg)
Blocking
b(2:8) bs(2:8)
11.90.92.41.40.91.63.4
11.93.42.41.40.90.91.6
![Page 34: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/34.jpg)
Qualitative factors
Design factors can be
Quantitative (continuous)
Qualitative (discrete)
→ Use of switch variables for discrete factors
E.g. effect of temperature and solvent (A, B or C) on extraction
where
1 if A isdiscrete level and 1 ifB isthediscrete level0 otherwise 0 otherwise
![Page 35: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/35.jpg)
Qualitative factors
![Page 36: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/36.jpg)
Session 3
Central composite designs
Design variance
Common designs
Second order models
Stationary points
ANOVA
Blocking
Confounding
Qualitative factors
![Page 37: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/37.jpg)
Nomenclature
Center-point
Axial point
Lack of fit
Prediction
Rotatability
Stationary point
Saddle point
Minimum
Maximum
Analysis of variance (ANOVA)
Response transformation
Blocking
Confounding
Qualitative factors
![Page 38: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/38.jpg)
Contents
Practical course, arranged in 4 individual sessions:
Session 1 – Introduction, factorial design, first order models
Session 2 – Matlab exercise: factorial design
Session 3 – Central composite designs, second order models, ANOVA,
blocking, qualitative factors
Session 4 – Matlab exercise: practical optimization example on given
data
![Page 39: S3 - Process product optimization design experiments response surface methodolgy - Session 3/4](https://reader034.vdocument.in/reader034/viewer/2022052623/559c1a151a28ab1d598b4741/html5/thumbnails/39.jpg)
Thank you for listening!
Please send me an email that you are attending the course