s3 - process product optimization design experiments response surface methodolgy - session 3/4
DESCRIPTION
Session 3/4 – Central composite designs, second order models, ANOVA, blocking, qualitative factors An intensive practical course mainly for PhD-students on the use of designs of experiments (DOE) and response surface methodology (RSM) for optimization problems. The course covers relevant background, nomenclature and general theory of DOE and RSM modelling for factorial and optimisation designs in addition to practical exercises in Matlab. Due to time limitations, the course concentrates on linear and quadratic models on the k≤3 design dimension. This course is an ideal starting point for every experimental engineering wanting to work effectively, extract maximal information and predict the future behaviour of their system. Mikko Mäkelä (DSc, Tech) is a postdoctoral fellow at the Swedish University of Agricultural Sciences in Umeå, Sweden and is currently visiting the Department of Chemical Engineering at the University of Alicante. He is working in close cooperation with Paul Geladi, Professor of Chemometrics, and using DOE and RSM for process optimization mainly for the valorization of industrial wastes in laboratory and pilot scales.” The course took place at the University of Alicante and would not had been possible without the support of the Instituto Universitario de Ingeniería de Procesos Químicos.TRANSCRIPT
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
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
Session 1
Introduction
Why experimental design
Factorial design
Design matrix
Model equation = coefficients
Residual
Response contour
Session 2
Factorial design
Research problem
Design matrix
Model equation = coefficients
Degrees of freedom
Predicted response
Residual
ANOVA
R2
Response contour
Session 3
Central composite designs
Design variance
Common designs
Second order models
Stationary points
ANOVA
Blocking
Confounding
Qualitative factors
Central composite designs
f(x) f(x)
x1 x2 x1 x2x3
First order f(x) Second order f(x)
Central composite designs
Second order models through
Center-points
Axial points
αnc
Central composite designs
Center-points (nc)
Pure error (lack of fit)
Curvature
Axial points (α)
Quadratic terms
Spherical designα > 1
Cuboidal designα = 1
Central composite designs
Design characteristics nc and α
Pure error (lack of fit)
Estimated error distribution
Area of operability
Control over factor levels
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
Central composite designs
CCD, k 2, 2, 5
CCD, k 2, 2, 1
Scaled prediction variance
Central composite designs
Common designs
Central composite α > 1
Central composite designs
Common designs
Central composite α = 1
Central composite designs
Common designs
Box-Behnken
Second order models
First order models
Main effects
Main effects + interactions
Second order models
Main effects + interactions + quadratic terms
⋯
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
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.
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
Research problem
Unrefined coefficients
Contour
Second order models
Second order models can include stationary points:
Saddle point Maximum/minimum
Second order models
Stationary point character can be described
Fitted second order model (k = 2)
Derivation 0 results in
2 0
2 0
Second order models
For analysing a stationary point
′ where
⋯ , ⋮ and
/2 ⋯ /2⋯ /2⋱ ⋮
sym.
→ location and character
Second order models
Stationary point location
From the previous example
0.5 0.2
485.8
Second order models
Stationary point character
/2 ⋯ /2⋯ /2⋱ ⋮
sym.
Eigenvalues
, , ⋯ , all < 0 → Maximum
, , ⋯ , all > 0 → Minimum
, , ⋯ , mixed in sign → Saddle point..
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
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
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
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.
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
ANOVA
Response transformations or modification of model terms might alleviate
lack of fit
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
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.
Blocking
b(2:8) bs(2:8)
11.90.92.41.40.91.63.4
11.93.42.41.40.90.91.6
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
Qualitative factors
Session 3
Central composite designs
Design variance
Common designs
Second order models
Stationary points
ANOVA
Blocking
Confounding
Qualitative factors
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
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
Thank you for listening!
Please send me an email that you are attending the course