RESPONSE SURFACE METHODOLOGY (R S M)

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RESPONSE SURFACE METHODOLOGY (R S M)

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Title: RESPONSE SURFACE METHODOLOGY (R S M)

1
RESPONSE SURFACE METHODOLOGY (R S M)

Par
Mariam MAHFOUZ

2
Remember thatGeneral Planning

Part I
A - Introduction to the RSM method
B - Techniques of the RSM method
C - Terminology
D - A review of the method of least squares
Part II
A - Procedure to determine optimum
conditions Steps of the RSM method
B Illustration of the method with an example

3

Part II
4
A - Procedure to determine optimum conditions
steps of the method

This method permits to find the settings of the
input variables which produce the most desirable
response values.
The set of values of the input variables which
result in the most desirable response values is
called the set of optimum conditions.

5
Steps of the method

The strategy in developing an empirical model
through a sequential program of experimentation
is as follows
The simplest polynomial model is fitted to a set
of data collected at the points of a first-order
design.
If the fitted first-order model is adequate, the
information provided by the fitted model is used
to locate areas in the experimental region, or
outside the experimental region, but within the
boundaries of the operability region, where more
desirable values of the response are suspected to
be.

In the new region, the cycle is repeated in that
the first-order model is fitted and testing for
adequacy of fit.
If nonlinearity in the surface shape is detected
through the test for lack of fit of the
first-order model, the model is upgraded by
adding cross-product terms and / or pure
quadratic terms to it. The first-order design is
likewise augmented with points to support the
fitting of the upgraded model.

If curvature of the surface is detected and a
fitted second-order model is found to be
appropriate, the second-order model is used to
map or describe the shape of the surface, through
a contour plot, in the experimental region.
If the optimal or most desirable response values
are found to be within the boundaries of the
experimental region, then locating the best
values as well as the settings of the input
variables that produce the best response values.

7. Finally, in the region where the most
desirable response values are suspected to be
found, additional experiments are performed to
verify that this is so.

9
B- Illustration of the method with an example

For simplicity of presentation we shall assume
that there is only one response variable to be
studied although in practice there can be several
response variables that are under investigation
simultaneously.

10
Experience
Two controlled Factors
Chemical reaction
One response
Temperature (X1)
percent yield
Time (X2)
An experimenter, interested in determining if an
increase in the percent yield is possible by
varying the levels of the two factors.
11
Two levels of temperature 70 and 90.
Two levels of time 30 sec and 90 sec.
Four different design points
Four temperature-time settings (factorial
combinations)
And two repetitions at each point
The total number of observations is N 8
12
Detail

The response of interest is the percent yield,
which is a measure of the purity of the end
product.
The process currently operates in a range of
percent purity between 55 and 75 , but it is
felt that a higher percent yield is possible.

13

Design 1 Design 1 Design 1 Design 1 Design 1
Original variables Original variables Coded variables Coded variables Percent yield
Temperature X1 (C) Time X2 (sec.) x1 x2 Y
70 30 -1 -1 49.8 48.1
90 30 1 -1 57.3 52.3
70 90 -1 1 65.7 69.4
90 90 1 1 73.1 77.8
x1 and x2 are the coded variables which are defined as x1 and x2 are the coded variables which are defined as x1 and x2 are the coded variables which are defined as x1 and x2 are the coded variables which are defined as x1 and x2 are the coded variables which are defined as

14
Representation of the first design
15
First-order model

Expressed in terms of the coded variables, the
observed percent yield values are modeled as
The remaining term, ?, represents random error
in the yield values.
The eight observed percent yield values, when
expressed as function of the levels of the coded
variables, in matrix notation, are
Y X ? ?

16
Matrix form
Vector of error terms
Vector of response values
Matrix of the design

Vector of unknown parameters
17
Estimations

The estimates of the coefficients in the
first-order model are found by solving the normal
equations
The estimates are
The fitted first-order model in the coded
variables is

18
ANOVA table design 1
Source Degrees of freedom d.f. Sum of squares SS Mean square F
Model 2 864.8125 432.4063 63.71
Residual 5 33.9363 6.7873
Lack of fit 1 2.1013 2.1013 0.264
Pure error 4 31.8350 7.9588

Test of adequacy
19
Individual tests of parameters

To do that the Student-test is used.
For the test of we have
And for we have
Each of the null hypotheses is rejected at the ?
0.05 level of significance owing to the
calculated values, 3.73 and 10.65, being greater
in absolute value than the tabled value,
T50.025 2.571.

20
Conclusion of the first analysis

The first order model is adequate.
That both temperature and time have an effect
on percent yield.
Since both b1 and b2 are positive, the
effects are positive.
Thus, by raising either the temperature or time
of reaction, this produced a significant
increase in percent yield.

21
Second stage of the sequential program

At this point, the experimenter quite naturally
might ask
If additional experiments can be performed
At what settings of temperature and time should
the additional experiments be run?
To answer this question, we enter the second
stage of our sequential program of
experimentation.

22
Contour plots

The fitted model
can now be used to map values of the estimated
response surface over the experimental region.
This response surface is a hyper-plane their
contour plots are lines in the experimental
region.
The contour lines are drawn by connecting two
points (coordinate settings of x1 and x2) in the
experimental region that produce the same value
of

In the figure above are shown the contour
lines of the estimated planar surface for percent
yield corresponding to values of 55, 60,
65 and 70 .

24
Performing experiments along the path of steepest
ascent

To describe the method of steepest ascent
mathematically, we begin by assuming the true
response surface can be approximated locally with
an equation of a hyper-plane
Data are collected from the points of a
first-order design and the data are used to
calculate the coefficient estimates to obtain the
fitted first-order model

Estimated response function
25

The next step is to move away from the center
of the design, a distance of r units, say, in the
direction of the maximum increase in the
response.
By choosing the center of the design in the
coded variable to be denoted by O(0, 0, , 0),
then movement away from the center r units is
equivalent to find the values of
which maximize
subject to the constraint
Maximization of the response function is
performed by using Lagrange multipliers. Let
where ? is the Lagrange multiplier.

26

27

To maximize subject to the
above-mentioned constraint, first we set equal to
zero the partial derivatives
i1,,k and
Setting the partial derivatives equal to zero
produces
i
1,,k, and
The solutions are the values of xi satisfying
or i 1,,k, where the
value of ? is yet to be determined. Thus the
proposed next value of xi is directly
proportional to the value of bi.

Let us the change in Xi be noted by ?i , and
the change in xi be noted by ?i. The coded
variables is obtained by these formulas
where
(respectively si) is the mean (respectively the
standard deviation) of the two levels of Xi .
Thus ,
then
or

Let us illustrate the procedure with the fitted
first-order model
that was fitted early to the percent yield
values in our example.
To the change in X2, ?245 sec. corresponds the
change in x2, ?245/301.5 units.
In the relation , we can
substitute ?i to xi
, thus
and ?1 0.526, so
?10.526105.3C .

30
Points along the path of steepest ascent and
observed percent yield values at the points

Temperature X1 (C) Time X2 (sec.) Observed percent yield
Base 80.0 60
?i 5.3 45
Base ?i 85.3 105 74.3
Base 1.5 ?i 87.95 127.5 78.6
Base 2 ?I 90.6 150 83.2
Base 3 ?I 95.9 195 84.7
Base 4 ?i 101.2 240 80.1
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Sequence of experimental trials performed in
moving to a region of high percent yield values

Design two For this design the coded
variables are defined as

x1 x2 X1 X2 yield
-1 -1 85.9 165 82.9 81.4
1 -1 105.9 165 87.4 89.5
-1 1 85.9 225 74.6 77.0
1 1 105.9 225 84.5 83.1
0 0 95.9 195 84.7 81.9
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The fitted model corresponding to the group of
experiments of design two is
The corresponding analysis of variance is

Source d.f. SS MS F
Model 2 162.745 81.372 42.34
Residual 7 13.455 1.922
Lack of fit 2 2.345 1.173 0.53
Pure error 5 11.110 2.222
Total (variations) 9 176.2
The model is jugged adequate
35
sequence of experimental trials that were
performed in the direction two

Steps x1 x2 X1 X2 yield
1 Center ?i 1 - 0.77 105.9 171.9 89.0
2 Center 2 ?i 2 - 1.54 115.9 148.8 90.2
3 Center 3 ?i 3 - 2.31 125.9 125.7 87.4
4 Center 4 ?i 4 - 3.08 135.9 102.6 82.6
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Retreat to center 2 ?i and proceed in
direction three

Steps x1 x2 X1 X2 yield
5 Replicated 2 2 - 1.54 115.9 148.8 91.0
6 3 - 0.77 125.9 171.9 93.6
7 4 0 135.9 195 96.2
8 5 0.77 145.9 218.1 92.9
38
Set up design three using points of steps 6, 7,
and 8 along with the following two points

Steps x1 x2 X1 X2 yield
9 3 0.77 125.9 218.1 91.7
10 5 - 0.77 145.9 171.9 92.5
11 Replicated 7 4 0 135.9 195 97.0

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Design three was set up using the point at step
7 as its center. It includes steps 6 11. If we
redefine the coded variables
and
then the fitted first-order model is
The corresponding analysis of variance table
is

41
ANOVA table
source d.f. SS MS F
Model 2 0.5650 0.2825 0.04
Residual 3 22.1833 7.3944
total 5 22.7483
It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface. It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface. It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface. It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface. It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface.

42
Fitting a second-order model

A second-order model in k variables is of the
form
The number of terms in the model above is
p(k1)(k2)/2 for example, when k2 then p6.
Let us return to the chemical reaction example
of the previous section. To fit a second-order
model (k2), we must perform some additional
experiments.

43
Central composite rotatable design

Suppose that four additional experiments are
performed, one at each of the axial settings
(x1,x2)
These four design settings along with the four
factorial settings (-1,-1) (-1,1) (1,-1)
(1,1) and center point comprise a central
composite rotatable design.
The percent yield values and the corresponding
nine design settings are listed in the table
below

44
Central composite rotatable design
45
Percent yield values at the nine points of a
central composite rotatable design

The fitted second-order model, in the coded
variables, is
The analysis is detailed in this table, using
the RSREG procedure in the SAS software

47
SAS output 1

48
SAS output 2
49
Response surface and the contour plot
50
More explanations

The contours of the response surface, showing
above, represent predicted yield values of 95.0
to 96.5 percent in steps of 0.5 percent.
The contours are elliptical and centered at the
point
(x1 x2)(- 0.0048 - 0.0857)
or (X1 X2)(135.85C 193.02 sec).
The coordinates of the centroid point are called
the coordinates of the stationary point.
From the contour plot we see that as one moves
away from the stationary point, by increasing or
decreasing the values of either temperature or
time, the predicted percent yield (response)
value decreases.

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Determining the coordinates of the stationary
point

A near stationary region is defined as a region
where the surface slopes (or gradients along the
variables axes) are small compared to the
estimate of experimental error.
The stationary point of a near stationary region
is the point at which the slope of the response
surface is zero when taken in all direction.
The coordinates of the stationary point
are calculated by differentiating the estimated
response equation with respect to each xi,
equating these derivatives to zero, and solving
the resulting k equations simultaneously.

Remember that the fitted second-order model in
k variables is
To obtain the coordinates of the stationary
point, let us write the above model using matrix
notation, as

where
and

55
Some details

The partial derivatives of with respect
to x1, x2, , xk are

56
More details

Setting each of the k derivatives equal to zero
and solving for the values of the xi, we find
that the coordinate of the stationary point are
the values of the elements of the kx1 vector x0
given by
At the stationary point, the predicted response
value, denoted by , is obtained by
substituting x0 for x

57
Return to our example

The fitted second-order model was
so the stationary point is
In the original variables, temperature and time
of the chemical reaction example, the setting at
the stationary point are temperature135.85C
and time193.02 sec.
And the predicted percent yield at the
stationary point is

58
Moore details

Note that the elements of the vector x0 do not
tell us anything about the nature of the surface
at the stationary point.
This nature can be a minimum, a maximum or a
mini_max point.
For each of these cases, we are assuming that
the stationary point is located inside the
experimental region.
When, on the other hand, the coordinates of the
stationary point are outside the experimental
region, then we might have encountered a rising
ridge system or a falling ridge system, or
possibly a stationary ridge.

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Nest Step

The next step is to turn our attention to
expressing the response system in canonical form
so as to be able to describe in greater detail
the nature of the response system in the
neighborhood of the stationary point.

61
The canonical Equation of a Second-Order Response
System

The first step in developing the canonical
equation for a k-variable system is to translate
the origine of the system from the center of the
design to the stationary point, that is, to move
from (x1,x2,,xk)(0,0,,0) to x0.
This is done by defining the intermediate
variables (z1,z2,,zk)(x1-x10,x2-x20,,xk-xk0)
or zx-x0.
Then the second-order response equation is
expressed in terms of the values of zi as

Now, to obtain the canonical form of the
predicted response equation, let us define a set
of variables w1,w2,,wk such that W(w1,w2,,wk)
is given by
where M is a kxk orthogonal matrix whose
columns are eigenvectors of the matrix B.
The matrix M has the effect of diagonalyzing B,
that is, where ?1,?2,,?k are the corresponding
eigenvalues of B.
The axes associated with the variables
w1,w2,,wk are called the principal axes of the
response system.
This transformation is a rotation of the zi
axes to form the wi axes.

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So we obtain the canonical equation
The eigenvalues ?i are real-valued (since the
matrix B is a real-valued, symmetric matrix) and
represent the coefficients of the terms in the
canonical equation.
It is easy to see that if ?1,?2,,?k are
1) All negative, then at x0 the surface is a
maximum.
2) All positive, then at x0 the surface is a
minimum.
3) Of mixed signs, that is, some are positive
and the
others are negative, then x0 is a saddle
point of
the fitted surface.
The canonical equation for the percent yield
surface is

65
Moore details

The magnitude of the individual values of the
?i tell how quickly the surface height changes
along the Wi axes as one moves away from x0.
Today there are computer software packages
available that perform the steps of locating the
coordinates of the stationary point, predict the
response at the stationary point, and compute the
eigenvalues and the eigenvectors.

For example, the solution for optimum response
generated from PROC RSREG of the Statistical
Analysis System (SAS) for the chemical reaction
data, is in following table

67
Recapitulate
Process to optimize
Contours and optimal direction

Input and output variables
Experiments in the Optimal direction
Experimental and Operational regions
Locate a new Experimental region
Series of experiments
New series of experiments
Yes
Fitting First-order model
Fitting First-order model
Model Adequate ?
Model Adequate ?
Yes
Fitting a Second-order model
No
No
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Doing this

Coordinates of the stationary point.
Description of the shape of the response surface
near the stationary point by contour plots.
Canonical analysis.
If needed, Ridge analysis (not detailed here).

69
Field of use of the method

In agriculture
In food industry
In pharmaceutical industry
In all kinds of the light and heavy industries
In medical domain
Etc.

70
Bibliography

André KHURI and John CORNELL Response Surfaces
Designs and Analyses , Dekker, Inc., ASQC
Quality Press, New York.
Irwin GUTTMAN Linear Models An Introduction,
John Wiley Sons, New York.
George BOX, William HUNTER J. Stuart HUNTER
Statistics for experimenters An Introduction to
Design, Data Analysis, and Model Building , John
Wiley Sons, New York.
George BOX Norman DRAPPER Empirical
Model-Building and Response Surfaces , John
Wiley Sons, New York.