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21/01/2014 • 1

Screening and Response surface: Concepts

Jean-Louis COULOMB

UMR CNRS 5269 - Grenoble-INP

– Université Joseph Fourier

January 2014

Screening - Response Surface

21/01/2014 Screening - Response Surface • 2

1. Introduction: Direct optimization

Direct optimization

Optimization algorithm controls the objective function directly.

Cost optimization depends linearly on the number of tests and their unit cost.

When the unit cost is important (long simulation or prototype building), there is a need to control the overall cost optimization.

Response Surface and Screening (parts of Design Of Experiments) address this need.

Optimization

algorithm

Objective

function

x

f(x)


1. Introduction: Indirect optimization

Indirect optimization

Optimization algorithm is applied to an approximation of the objective function.

This approximation is called Response Surface.

The response surface can be constructed a priori (before optimization) or during the optimization process (then it can be adaptive).

The cost of the calls to the approximation is negligible. Only the number of evaluations of the objective function account.

Optimization

algorithm

Response

surface

x

f ap (x)

Objective

function

x grid

f( x grid )

Numerous

inexpensive

calls

Few

expensive

calls


1. Introduction: Screening

Screening

The screening is based on a simple response surface (polynomials of order 1 or 2).

The coefficients of variables and interactions are identified by means of a minimum number of tests

The most important coefficients are the mark of the most significant variables and interactions.

Not significant variables can be fixed which significantly reduces the optimization cost.

Screening

x 1 , x 2 , x 3 , … x 1 x 2 , x 2 x 3 , …

Direct

optimization

Indirect

optimization

x 1 , x 3


Summary

1. Introduction

2. Design Of Experiments

3. Response Surface

4. Design Of Experiments Practical work – Role playing


2. Design Of Experiments: History

1925: Agricultural research (Fisher) An area where experiments are long, expensive and the results are subject to variability.

It was a method to get from a given number of trials, a maximum credible information on the influence of factors and their prioritization.

1945-1960: Enrichment by statisticians (Box, Hunter) They introduced fractional plans at two levels, central composite designs and associated response surface models.

1960: Quality Approach (Taguchi) Build quality upstream from the design.

Design efficient products on average with small variation around this average.

Made efficiency less sensitive to conditions of use, to manufacturing variations and aging (robust design).


2. Design Of Experiments: Surface response: Example

Find a Surface Response for reaction efficiency Y as a function of three factors each with two levels

The study is conducted sequentially in three stages to minimize the number of experiments:

Step 1: Estimate the variability of the response Y.

Step 2: Postulate for Y a linear model without interaction.

Step 3 (if step 2 fails): Postulate for Y a linear model with interactions.

Factor Meaning Lower level Upper level Normalized - Normalized +

X1 Temperature (℃) 100 200 x1= 1 x1= +1

X2 Pressure (bar) 1 2 x2= 1 x2= +1

X3 Catalyser Product 1 Product 2 x3= 1 x3= +1


2. Design Of Experiments: Example Step 1

Step 1: Estimate the variability of the response Y Typically, this estimation is achieved by repetition of the experiment at the center of the domain. Here, this is impossible because the factor X3 is a discrete factor.

We choose to repeat 3 experiments X1=150℃ X2=1.5 bars X3=Product 2

The 3 answers are Y1=40.3% Y2=39.8% Y3=40.9%

The average is = 40.33%

The standard deviation is = 0.55%

The experimental error is quite low => a DOE is possible

Note: When results come from simulations => there is no variability

3213

1YYYY

2

3

2

2

2

113

1' YYYYYY


2. Design Of Experiments: Example Step 2 (strategy)

Step 2: Postulate for Y a linear model without interaction The model is

4 unknown coefficients (a0, a1, a2, a3) => 4 experiments

3322110 xaxaxaaY

Optimal strategy => Orthogonality + + +

+

+

+

X1 at level =>

X2 at level =>

etc

X2 1 time et 1 time +




N° experiment Factor X1 (°C) Factor X2 (bar) Factor X3 Response Y (%)

1 200 2 Product 2 75

2 100 2 Product 1 56

3 200 1 Product 1 14

4 100 1 Product 2 9


2. Design Of Experiments: Example Step 2 (test)

Experiments

Resulting model (least square))

Test the validity of the model (at combination unused of levels + and -)

Ypredict=2% very different from Yexp=10% => Change model for taking into account interactions

321 5.32765.38 xxxY

N° experiment Factor X1 (°C) Factor X2 (bar) Factor X3 Réponse Y (%)

5 100 1 Product 1 10


2. Design Of Experiments: Example Step 3 (strategy)

Step 3: Postulate for Y a linear model with interactions Model

8 unknown coefficients => 8 experiments

Optimal strategy => Full factorial design at 2 levels

5 tests are already available

• 4 for building the previous model

• +1 for testing

Only three additional experiments to perform (sequentiality)!

3211233223311321123322110 xxxaxxaxxaxxaxaxaxaaY

N° experiment Factor X1 (°C) Factor X2 (bar) Factor X3 Response Y (%)

6 200 2 Product 1 66

7 200 1 Product 2 34

8 100 2 Product 2 45


2. Design Of Experiments: Example Step 3 (test)

Resulting model

The absolute value of a coefficient given the impact of the linear effect or interaction.

The interaction x1x2x3 is negligible => refined model

Test the validity of the model Central point of start of the study (can ensure absence of non-linearity) X1=150°C, X2=1.5 bars, X3= Product 2 Responses are Y1=40.3% Y2=39.8% Y3=40.9% => Ypredict = 40.73%

Small difference => this linear model with interactions seems acceptable

We ignore here the statistical validation which is an important part of the method for real experiments (not for numerical)

Usable for interpolation only (NOT for extrapolation)

321323121321 13.063.213.538.113.288.2163.863.38 xxxxxxxxxxxxY

323121321 63.213.538.113.288.2163.863.38 xxxxxxxxxY


2. Design Of Experiments: Methodology in brief

Define N factors and the experimental region of interest

Define the normalized variables associated with variable factors

Step 1: Apply a non-linear interaction model

Conduct experiments

Determine the linear model without interaction

Validate the model

Exploitation of the linear model without interaction

Step 2 (if previous model not valid): Apply a linear model with interactions

Conduct experiments

Determine the linear model with interactions

Validate the model (center field)

Exploitation of the linear model with interaction

Step 3 (if previous model not valid): Apply a second degree model ...


2. Design Of Experiments: Fractional factorial designs

Now we have more factors, but hopefully we know a “good” model

We know that the model depends on 4 factors and only 3 of first order interactions

If we implement a 24 factorial design, it will take 16 experiences while 8 would be sufficient!

Idea

Take a subset of the 24 factorial design that respects the orthogonality between factors.

There are tables that provide designs based on the number of factors and the number of experiments (power of 2) desired (Box, Hunter and Hunter).

Note

We do not calculate the effects (a1, a13, ...) but additions of effects (a1+a234, a13+a24, ...) called contrasts (we say that a1 is an alias for a234 ...).

This introduces confusions that depend on the DOE and on the order of the factors in this DOE.

Using known a priori information on significant interactions (a234=a24= ... =0), the experimenter can choose a design such that the remaining effects (a1, a13, ...) are calculated.

433432233113443322110 xxaxxaxxaxaxaxaxaaY


2. Design Of Experiments: Summary

Design of experiments for the experimenter are a very effective way

To determine the influential factors of a system (screening)

To predict the response of a system (response surface)

Optimize system (at least second degree response surface)

The analysis of the variability of response (not shown in the introduction) allows

To build quality upstream from the design

Design efficient products on average with small variation around this average

To make the efficiency less sensitive to conditions of use, to manufacturing variations and aging (robust design)

For the numerical DOE we retain The method of screening before optimization to eliminate non-influential factors

The response surface method in place of direct calculation of the objective function


3. Response Surface: Classical Response Surfaces

Classical Response Surfaces (1st and 2nd order polynomial) Polynomial of degree 1 without interaction

Polynomial of degree 1 with interactions

Polynomial of degree 2

normalized variables,

the center value, and coefficients of the main effects and the coefficients of interactions.

These unknown coefficients can be determined by the least squares method

At least as many experiments of the objective function as unknown coefficients

Some additional experiments allow to test the response surface

Allow screening absolute value of a coefficient = impact of an effect

n

i

ii xaaxf1

0)(

1

1 11

0)(n

i

n

ij

jiij

n

i

ii xxaxaaxf

n

i

iii

n

i

n

ij

jiij

n

i

ii xaxxaxaaxf1

21

1 11

0)(

ix

0a ia iiaija


3. Response Surface: 1st order without interaction

Which experiments to choose? Fractional factorial designs at two levels

Nb

variables

Response Surface Nb

coefficients

1 110)( xaaxf 2

2 222110)( xaxaaxf 3

3 3322110)( xaxaxaaxf 4

… … …

n

n

iii xaaxf

10)(

1+n


3. Response Surface: 1st order with first interactions

Which experiments to choose? Fractional factorial designs at two levels

Nb

variables

Response Surface Nb

coefficients

1 110)( xaaxf 2

2 211222110)( xxaxaxaaxf 4

3 3113322321123322110)( xxaxxaxxaxaxaxaaxf 7

… … …

n

1

1 110)(

n

i

n

ijjiij

n

iii xxaxaaxf

2

)1.(1

nn


3. Response Surface: 2nd order (1/3)

Which experiments to choose?

Nb

variables

Response Surface Nb

coefficients

1 2

111110)( xaxaaxf 3

2 2

222

2

111211222110)( xaxaxxaxaxaaxf 6

3 2

3133

2

222

2

1110 ...)( xaxaxaaxf 10

… … …

n

n

iiii

n

i

n

ijjiij

n

iii xaxxaxaaxf

1

21

1 110)(

2

)3.(1

nn



The 2n factorial designs (2 levels) do not determine the quadratic effects

Example: => 2 trials (x1=-1, x1=+1) for 3 coefficients!

Beyond 3 factors, 3n factorial designs (3 levels) are too expensive

When in a study all the factors are continuous, the central composite designs of Box and Wilson are cheaper

2

111110 xaxaa

Number of factors n 2 3 4 5 6

Number of coefficients in 2nd order model = 2

)3.(1

nn

6 10 15 21 28

Number of experiments with 3n factorial design 9 27 81 243 729

Number of factors n 2 3 4 5 6

Number of experiments of the central composite design

(for only 1 central point)

9 15 25 27 45

= Factoriel design 2n ou 2n-1 + 2n points on axis

+ Nc points on center c

n Nn 22 c

n Nn 22 1



Spherical central composite design

5 levels (, 1, 0, +1, +)

For n=3 : =1.682 and Nc=6

Tradeoff between

=> quasi-orthogonality

=> isovariance by rotation

=> uniform accuracy

Cubical central composite design

Only 3 levels (1, 0, +1)

Simpler


3. Response Surface: Classical Response Surfaces

Advantages and drawbacks of classical response surfaces

Advantages :

They allow the examination of the main

effects of factors and their interactions.

This can be used to exclude the least

significant factors before optimization

(screening).

Function with three local minima and a global

minimum

Drawbacks :

These response surfaces cannot replace a

multimodal function (with multiple optima).


3. Response Surface: Generalised Response Surfaces

Aim Provide a correct approximation method for multimodal function

Principles common to generalized response surface Grid points M (possibly refined around the optimum)

Approximation of the form

Method of radial basis functions Inverse multiquadric

Gaussian

M

j

jj xxhcxf1

2jxx

j exxh


4. Design Of experiments: Pratical work

Experiment “Design Of Experiments” Document “Role playing”

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