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**DOE: Optimization Response Surface Methods**

6BV04 DOE: Optimization Response Surface Methods

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**Contents Optimisation steps Box method Steepest ascent method**

Practical example Response surface designs Multiple responses EVOP Software Literature

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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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optimum improvement current settings

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**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 = ß0 + ß1x1 + ß2x2 + “Near” the optimum often a quadratic (second order) model suffices. For example: Y = ß0 + ß1x1 + ß2x2 + ß12x1x2 + ß11x12 + ß22x22 + 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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Models Far away from optimum: first order model

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**fitting a first order model**

Models Near optimum: fitting a first order model shows lack-of-fit (curvature)

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Models Near optimum: second order model

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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 1st 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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**Box method direction of largest increase 41.2 41.3 40.6 41.9 41.8**

39.3 40.9 41.5 40.0 stop if one has to return to previous settings

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**Steepest ascent method**

perpendicular to contour line direction of steepest ascent contour lines of first-order model region where 1eorder-model has been determined

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**full factorial + centre points**

Optimization scheme start end screening accept stationary point no yes stationary point optimum? stationary point nearby? 1st order model OK? no full factorial + centre points yes RSM design (CCD, ...) fit 2nd order model Bij Statlab: als stationair punt te ver weg is, opnieuw 1e orde model fitten, dus geen 2e orde model bij het geschatte optimum. yes no single observation in direction steepest ascent go to stationary point yes no better observation?

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**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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**Steepest ascent 22-design with 5 centre points:**

time: min; temp: °C results: montgomery14-1.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. Interactie plot maken Lack of fit toets doen

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**Steepest ascent path direction path: normal vector**

outcome analysis of measurement: yield = *time *temp with coding: x1= (time-35)/5 x2 = (temp-155)/5 yield = *x *x2 direction path: normal vector De vector van steepest ascent is altijd de vector van regressie-coefficienten. Laat zien dat Statgraphics ook via table options het path van steepest ascent kan laten zien!!! step size: 5 min reaction time (choice of chemical engineer!) coded step size temp (= 2.1°C)

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**Steepest ascent path experiments**

Further experiments with factor settings of experiment nr. 10.

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**Near the optimum Settings experiment 10: time = 85 min**

temperature = 175 °C A 22 design with 5 centre points is executed. results: montgomery14-4.sfx Lack-of-fit indicates curvature. Hence, we now are probably near the optimum.

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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 2p-design with centre points does not suffice, because then all quadratic factors are confounded. A 3p-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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**Response surface designs**

The following designs are widely used for fitting a quadratic model: Central Composite Design (uniform precision of effect estimates) Box-Behnken Design (almost uniform precision of effect estimates, but usually fewer runs required than for CCD) 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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**Central Composite Design**

A CCD consists of 3 parts: factorial points centre points axial points A CCD is often executed by adding points to an already performed 2p-design (highly efficient, but beware of blocking!).

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**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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**Box-Behnken designs These are designs that consists of**

combinations from 2p-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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**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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**Recap: optimisation in dimension1**

necessary condition for extremum: 1st derivative = 0 not sufficient: “point of inflection” extra sufficient condition: 2nd derivative 0

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**Zero first derivatives: saddlepoint vs. maximum**

maximum (favourable) saddle point (unfavourable)

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**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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**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 l’s are sometimes called “parameters of canonical form”)

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**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.sfx

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**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 = / BB = / Standard errors are based on total error with 7 d.f.

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**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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**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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**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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**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 = / BB = / Standard errors are based on total error with 7 d.f.

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**Stationarity and matrix analysis**

In Matlab: create matrix B and vector b compute eigenvalues and location of stationary point >> b = [ ; ] /2 b = 0.9950 0.5152

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**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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**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 = 0.3893 0.3059 < (distance star point) → inside experimental region

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**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]

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**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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**full factorial + centre points**

Optimization scheme start end screening accept stationary point no yes stationary point optimum? stationary point nearby? 1st order model OK? no full factorial + centre points yes RSM design (CCD, ...) fit 2nd order model Bij Statlab: als stationair punt te ver weg is, opnieuw 1e orde model fitten, dus geen 2e orde model bij het geschatte optimum. yes no single observation in direction steepest ascent go to stationary point yes no better observation?

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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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**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 2k-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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**Software design experiment with pre-defined catalogue**

StatLab optimisation: Interactive software for teaching DOE through cases Box: Game-like demonstration of Box method 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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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: DOE booklet from Umetrics: Introduction to DOE from moresteam.com StatSoft Electronic Statistics Textbook, chapter on experimental design NIST Engineering Statistics Handbook:

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