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/ department of mathematics and computer science BV04 DOE: Optimization Response Surface Methods

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

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/ department of mathematics and computer science Optimisation steps Optimisation is achieved by going through the following phases: screening (determine which factors really influence the outcome; tool: screening designs like fractional factorial) improvement (approach optimum by repeated change of factor settings; tools: Box/simplex or steepest ascent approach) determination of optimum (find optimal settings of factor settings; tool: response surface designs like CCD or Box-Behnken + analysis of response surface using eigenvalues)

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/ department of mathematics and computer science current settings improvement optimum

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/ department of mathematics and computer science Regression models used in optimisation Statistical techniques for optimisation assume the following (often reasonably satisfied in practice): “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 Lack-of-fit techniques must be applied in order to check whether these models are appropriate, since we cannot directly see whether we are near the optimum (cf. next slides).

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

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

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

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/ department of mathematics and computer science Improvement In order to efficiently move from current factor settings to factor setting that yield near-optimal values, 2 methods are available: Box/Simplex method –idea: form new full factorials in direction of largest increase in current full factorial –simple; no statistics needed for implementation –not efficient Steepest ascent/descent method –idea: use 1 st order regression model from fractional factorial to obtain direction of largest increase (“steepest ascent”) –perform single runs in direction of largest increase until increase stops –advanced –recommended since it is more efficient

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/ 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

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/ 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

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/ 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

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

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/ 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.

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/ 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)

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

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/ department of mathematics and computer science Near the optimum 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 Lack-of-fit indicates curvature. Hence, we now are probably near the optimum.

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/ department of mathematics and computer science Quadratic models In order to fit a quadratic model (suitable when we are near the optimum), 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.

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/ department of mathematics and computer science Response surface designs The following designs are widely used for fitting a quadratic model: Central Composite Design (uniform precision of effect estimates)Central Composite Design Box-Behnken Design (almost uniform precision of effect estimates, but usually fewer runs required than for CCD)Box-Behnken Design The choice between these models is usually decided by the availability of these designs for a given number of runs and number of factors. Note that there are other suitable designs (usually available in statistical software that supports DOE).

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/ 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 (highly efficient, but beware of blocking!).

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/ 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. In other words, one has the same precision for all factor estimates.

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/ 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 extreme conditions which are often hard to set)

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/ 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 using a response surface design (CCD or Box-Behnken)? The next slides show the tools to derive optimal settings and the pitfalls that have to be avoided.

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/ department of mathematics and computer science Recap: optimisation in dimension1 necessary condition for extremum: 1 st derivative = 0 not sufficient: “point of inflection” extra sufficient condition: 2 nd derivative 0

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/ department of mathematics and computer science Zero first derivatives: saddlepoint vs. maximum saddle point (unfavourable) maximum (favourable)

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

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/ department of mathematics and computer science Stationarity and matrix analysis stationary point (zero first-order derivatives): 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”)

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/ 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

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/ department of mathematics and computer science Stationarity and matrix analysis Use Matlab to avoid manual computations: 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.

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/ 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 =

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/ 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)

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/ 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

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/ 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.

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/ 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 =

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/ 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

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/ 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

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/ 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

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/ 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 =

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/ 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

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/ 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.

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/ department of mathematics and computer science Evolutionary Operation (EVOP) Optimisation of a running production process is not always possible or may not be allowed because of costs: involves interruption may (temporarily) yield low quality products An alternative is Evolutionary Operation: experimentation within running operation frequent execution of 2 k -designs, starting at current settings high and low setting of factors are close to each other, thus no risk of low quality products

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/ department of mathematics and computer science Software StatLab optimisation: Interactive software for teaching DOE through cases Box: Game-like demonstration of Box methodhttp://www.win.tue.nl/~marko/box/box.html Matlab virtual reactor: Statistics toolbox -> Demos -> Empirical Modeling -> RSM demo Statgraphics: menu choice Special -> Experimental Design –design experiment with pre-defined catalogue –analysis of experiments with ANOVA

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/ 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 DOE booklet from Umetrics: Introduction to DOE from moresteam.com StatSoft Electronic Statistics Textbook, chapter on experimental designchapter on experimental design NIST Engineering Statistics Handbook:

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