Process/Product Optimization Using Design of Experiments and Response Surface Methodology

Process/product optimization using design of experiments and response surface methodology

Mikko Mäkelä Sveriges landbruksuniversitet Swedish University of Agricultural Sciences Department of Forest Biomaterials and Technology Division of Biomass Technology and Chemistry Umeå, 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

First order f(x) Second order f(x)

f(x) f(x)

x x x1 x2 1 x3 2 Central composite designs

Second order models through

. Center-points nc α . Axial points Central composite designs

Center-points (nc) Spherical design . Pure error (lack of fit) α > 1 . Curvature

Axial points (α) . Quadratic terms Cuboidal design α = 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

Scaled prediction variance (SPV):

NVar x SPV Practical design optimality σ

. Model parameters (βi) SPV = f(r) . Prediction () quality

r Prediction () quality emphasized

. Design rotatability [0, 0] Central composite designs

Scaled prediction variance

CCD, k2, 2, 5

CCD, k2, 2, 1 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 1-1-11 1 2 1 -1 -1 1 Factorial 3-11-1 1 4111 1 Design matrix, k = 2 5-α 00 0 6 α 00 0 Axial 70-α 0 α2 80α 0 α2 9000 0

Center-points 10 0 0 0 0 11 0 0 0 0 Research problem

A central composite design was Measured response, y performed for a tire tread compound . Tire abrasion index

. Two factors x1 and x2 . Axial distance α = 1.633

. N:o of center-point nc = 4

Factor Factor levels

x1 -1.633 -1 0 1 1.633

x2 -1.633 -1 0 1 1.633

Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 275. Research problem

N:o x1 x2 x12 x11 x22 y

1-1-1111270 2 1 -1 -1 1 1 270 Factorial 3 -1 1 -1 1 1 310 411111240 5 -1.633 0 0 2.667 0 550 6 1.633 0 0 2.667 0 260 Axial 7 0 -1.633 0 0 2.667 520 8 0 1.633 0 0 2.667 380 900000520 1000000290 Center-points 1100000580 1200000590 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

/2 ⋯ /2 ⋯/2 ⋯ , and ⋮ ⋱⋮ 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 . ANOVA

Coefficients . Response dependent of a coefficient

. H0: ⋯ 0

. H1: 0for 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

Sum of Mean Parameter df F-value p-value squares (SS) square (MS) Total corrected n-1 SStot MStot MSmod <0.05 Regression k SSmod MSmod /MSres >0.05 Residual n-p SSres MSres n-p- MSlof/ <0.05 Lack of fit (n -1) SSlof MSlof c MSpe >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 2 . Statistically significant coefficients x1, x2, x3 and x1 . 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

Sum of Mean Parameter df F-value p-value squares (SS) square (MS)

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

3 . In a 2 design with 2 blocks, confound nuisance to x123 Blocking

E.g. 2 blocks based on the x123 interaction (randomized within blocks)

N:o x1 x2 x3 x123 y 1----90 2+- -+64 3-+-+81 4++- -63 5--++77 6+-+-61 7-++-88 8++++53

Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 126. Blocking

b(2:8) bs(2:8)

11.9 11.9 0.9 3.4 2.4 2.4 1.4 1.4 0.9 0.9 1.6 0.9 3.4 1.6 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

1ifA is discrete level 1if B is the discrete level and 0 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 Analysis of variance (ANOVA) Axial point Response transformation Lack of fit Blocking Prediction Confounding Rotatability Qualitative factors Stationary point Saddle point Minimum Maximum Contents

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