/ department of mathematics and computer science DS01 Statistics 2 for Chemical Engineering lecture 3

/ department of mathematics and computer science Contents Optimisation steps Box method Steepest ascent method Practical example Response surface designs Multiple responses EVOP Software Literature

/ department of mathematics and computer science Optimisation steps Optimisation is achieved by going through the following phases: screening (which factors are of importance) improvement (approach optimum as fast as possible: steepest ascent/ simplex) determination of optimum (response surface designs)

/ department of mathematics and computer science current settings improvement optimum

/ department of mathematics and computer science Models Far away from the optimum a first order model often suffices. for example: Y = ß 0 + ß 1 x 1 + ß 2 x 2 +  Near the optimum often a quadratic (second order) model suffices. For example: Y = ß 0 + ß 1 x 1 + ß 2 x 2 + ß 12 x 1 x 2 + ß 11 x ß 22 x 

/ department of mathematics and computer science Models Far away from optimum: first order model

/ department of mathematics and computer science Models Near optimum: fitting a first order model shows lack-of- fit (curvature)

/ department of mathematics and computer science Models Near optimum: second order model

/ department of mathematics and computer science Improvement In order to quickly find factor settings which yield near- optimal values, 2 methods are available: Box method (simple) Steepest ascent/descent method (advanced)

/ department of mathematics and computer science Box method direction of largest increase direction of largest increase stop if one has to return to previous settings

/ department of mathematics and computer science Steepest ascent method direction of steepest ascent contour lines of first order model perpendicular to contour line region where 1 e order-model has been determined

/ department of mathematics and computer science Practical example goal: maximise yield of chemical reactor significant factors: reaction time reaction temperature current factor setting: time = 35 min. temp = 155 °C current yield: 40 %

/ department of mathematics and computer science Steepest ascent 2 2 -design with 5 centre points: time: min; temp: °C results: montgomery14-1.sfxmontgomery14-1.sfx there is no significant interaction there is no significant lack-of-fit the regression model is significant Hence, we are not near the optimum.

/ department of mathematics and computer science Steepest ascent path outcome analysis of measurement: yield = *time *temp with coding: x 1 = (time-35)/5 x 2 = (temp-155)/5 yield = *x *x 2 direction path: normal vector step size: 5 min reaction time (choice of chemical engineer!) coded step size temp (= 2.1 °C)

/ department of mathematics and computer science Steepest ascent path experiments Further experiments with factor settings of experiment nr. 10.

/ department of mathematics and computer science Settings experiment 10: time = 85 min temperature = 175 °C A 2 2 design with 5 centre points is executed. results: montgomery14-4.sfxmontgomery14-4.sfx Now we are probably near the optimum.

/ department of mathematics and computer science Quadratic models In order to fit a quadratic model, we must vary the factors at 3 levels. A 2 p -design with centre points does not suffice,because then all quadratic factors are confounded. A 3 p -design is possible, but not to be recommended: number of runs grows fast uses more runs than necessary to fit quadratic model.

/ department of mathematics and computer science Response surface designs The following designs are widely used for fitting a quadratic model: Central Composite Design Box-Behnken Design However, there are other suitable designs.

/ department of mathematics and computer science Central Composite Design A CCD consists of 3 parts: factorial points axial points centre points A CCD is often executed by adding points to an already performed 2 p -design (but beware of blocking!).

/ department of mathematics and computer science Rotatability In a CCD there are 2 possible choices: number of centre points location axial points By choosing the axial points at the locations ( ,0,…,0) etc. with  = (# factorial points) ¼, the design becomes rotatable, i.e. the precision (variance) of the model depends on the distance to the origin only.

/ department of mathematics and computer science Box-Behnken designs These are designs that consists of combinations from 2 p -designs. Properties: efficient (few runs) (almost) rotatable no corner points of hypercube (these are often extreme conditions and hence often hard to set)

/ department of mathematics and computer science Stationary point Near the optimum usually a quadratic model suffices: How do find the optimum after we correctly estimated the parameters?

/ department of mathematics and computer science One-dimensional case necessary condition for extremum: 1 st derivative = 0 not sufficient: “point of inflection” extra sufficient condition: 2 nd derivative  0

/ department of mathematics and computer science Saddlepoint vs. maximum saddle point maximum

/ department of mathematics and computer science Analysis response surface model Graphically: make contourplot (if 2 factors) Analytically: matrix notation: Note: B must be chosen as symmetric matrix, see example:

/ department of mathematics and computer science Stationarity and matrix analysis stationary point: characterisation through eigenvalues of matrix B: all eigenvalues positive: min all eigenvalues negative: max eigenvalues different signs: saddle point (the ’s are sometimes called “parameters of canonical form”)

/ department of mathematics and computer science Stationarity and matrix analysis In StatGraphics: augment design add Star points Please note that additional centre points are added and a block variable. We can remove the centre points from the data set and ignore the block variable in the analysis. StatGraphics results: montgomery14-6.sfxmontgomery14-6.sfx

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> B = [ /2 ; 0.5/ ] / 2 Analysis Summary File name: D:\MyDocs\2DS01\collegesheets\montgomery14-6.sfx Comment: Montgomery table 14-4 Estimated effects for opbrengst average = / A:tijd = / B:temperatuur = / AA = / AB = 0.5 +/ BB = / Standard errors are based on total error with 7 d.f.

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> B = [ /2 ; 0.5/ ] / 2 B =

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> eig(B)

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> eig(B) ans = both negative → maximum

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> b = [ ; ] /2 Analysis Summary File name: D:\MyDocs\2DS01\collegesheets\montgomery14-6.sfx Comment: Montgomery table 14-4 Estimated effects for opbrengst average = / A:tijd = / B:temperatuur = / AA = / AB = 0.5 +/ BB = / Standard errors are based on total error with 7 d.f.

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> b = [ ; ] /2 b =

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> spcoded = -0.5 * inv(B) * b

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> spcoded = -0.5 * inv(B) * b spcoded = < (distance star point) → inside experimental region

/ department of mathematics and computer science In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> sporiginal = spcoded.* [5 ; 5] + [85 ; 175] Stationarity and matrix analysis

/ department of mathematics and computer science Stationarity and matrix analysis In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> sporiginal = spcoded.* [5 ; 5] + [85 ; 175] sporiginal =

/ department of mathematics and computer science start screening RSM design (CCD,...) single observation in direction steepest ascent full factorial + centre points 1 st order model OK? better observation? stationary point optimum? stationary point nearby? go to stationary point yes no yes no yes no accept stationary point end fit 2 nd order model yes Optimization scheme

/ department of mathematics and computer science Multiple responses If more than 1 response variable needs to be optimised, then a graphical way of optimising may be achieved by overlaying contour plots in case there are only 2 independent variables.

/ department of mathematics and computer science Evolutionary Operation (EVOP) Optimisation of a running production process is costly: involves interruption may (temporarily) yield low quality products An alternative is Evolutionary Operation: frequent execution of 2 k -designs high and low setting of factors are close to each other

/ department of mathematics and computer science Software StatLab optimisation: Interactive software for teaching DOE through cases Box: Game-like demonstration of Box methodhttp:// Matlab virtual reactor: Statistics toolbox -> Demos -> Empirical Modeling - > RSM demo Better beer brewery DOE simulation (with extra chemical information): chemical information Statgraphics: menu choice Special -> Experimental Design –design experiment with pre-defined catalogue –analysis of experiments with ANOVA

/ department of mathematics and computer science Literature J. Trygg and S. Wold. Introduction to Experimental Design – What is it? Why and Where is it Useful?, Homepage of Chemometrics, editorial August 2002: 02.html 02.html StatSoft Electronic Statistics Textbook, chapter on experimental designchapter on experimental design NIST Engineering Statistics Handbook: