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 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 [email protected].