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 R 2 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 1 x 2 x 1 x 3 x 2
Central composite designs Second order models through Center-points Axial points n c α
Central composite designs Center-points (n c ) Pure error (lack of fit) Curvature Spherical design α > 1 Axial points (α) Quadratic terms Cuboidal design α = 1
Central composite designs Design characteristics n c and α Pure error (lack of fit) Estimated error distribution Area of operability Control over factor levels
Central composite designs Scaled prediction variance (SPV): Practical design optimality Model parameters (β i ) Prediction ( ) quality SPV NVar x σ SPV = f(r) Prediction ( ) quality emphasized Design rotatability [0, 0] r
Central composite designs Scaled prediction variance CCD, k 2, 2, 5 CCD, k 2, 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 Factorial Design matrix, k = 2 Axial Center-points N:o x i x j x ij x ii x jj 1-1 -1 1 1 2 1-1 -1 1 3-1 1-1 1 4 1 1 1 1 5 -α 0 0 0 6 α 0 0 0 7 0 -α 0 α 2 8 0 α 0 α 2 9 0 0 0 0 10 0 0 0 0 11 0 0 0 0
Research problem A central composite design was performed for a tire tread compound Measured response, y Tire abrasion index Two factors x 1 and x 2 Axial distance α = 1.633 N:o of center-point n c = 4 Factor Factor levels x 1-1.633-1 0 1 1.633 x 2-1.633-1 0 1 1.633 Myers, Montgomery & Anderson-Cook, Response Surface Methodology, 3rd ed., 2009, 275.
Research problem N:o x 1 x 2 x 12 x 11 x 22 y Factorial Axial Center-points 1-1 -1 1 1 1 270 2 1-1 -1 1 1 270 3-1 1-1 1 1 310 4 1 1 1 1 1 240 5-1.633 0 0 2.667 0 550 6 1.633 0 0 2.667 0 260 7 0-1.633 0 0 2.667 520 8 0 1.633 0 0 2.667 380 9 0 0 0 0 0 520 10 0 0 0 0 0 290 11 0 0 0 0 0 580 12 0 0 0 0 0 590
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..
ANOVA Coefficients Response dependent of a coefficient H 0 : 0 H 1 : 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 (H 0 ) Ratio of variances and respective dfs Distribution for every combination of dfs One- or two-tailed Alternative hypothesis (H 1 ) upper one-tailed (reject H 0 if F F,df,df )
ANOVA Parameter df Sum of squares (SS) Mean square (MS) F-value p-value Total corrected n-1 SStot MStot Regression k SSmod MSmod Residual n-p SSres MSres MSmod /MSres <0.05 >0.05 Lack of fit n-p- (n c -1) SSlof MSlof MSlof/ MSpe <0.05 >0.05 Pure error n c -1 SSpe MSpe p = k + 1 MS = SS / df
Research problem An extraction process (x 1,x 2,x 3 ) was studied using a cuboidal central composite design (α = 1, n c = 3) for maximizing yield Statistically significant coefficients x 1, x 2, x 3 and x 2 1 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) Mean square (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 2 k design in 2 p blocks where p < k In a 2 3 design with 2 blocks, confound nuisance to x 123
Blocking E.g. 2 blocks based on the x 123 interaction (randomized within blocks) N:o x 1 x 2 x 3 x 123 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 0.9 2.4 1.4 0.9 1.6 3.4 11.9 3.4 2.4 1.4 0.9 0.9 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 1 if A is discrete level and 1 if B is the discrete level 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 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 mikko.makela@slu.se