Special Topics
Example 1  Simultaneous Optimization of Several Response Variables in
a Central Composite (Response Surface) Design
Overview. This
example describes how to profile predicted responses and response desirability
using the Response/Desirability
Profiler. The procedures used in product development generally involve
two steps: 1) predicting responses on the dependent, or Y,
variables by fitting the observed characteristics of the product using
an equation based on the levels of the independent, or X,
variables, and 2) finding the levels of the X
variables that simultaneously produce the most desirable predicted responses
on the Y variables.
Derringer and Suich (1980) give, as an example of these procedures,
the problem of finding the most desirable tire tread compound. In their
example, there are four Y variables:
PICO Abrasion Index, 200 percent modulus, elongation at break, and hardness.
The characteristics of the product in terms of the response variables
depend on the ingredients, the X
variables: hydrated silica level, silane coupling level, and sulfur. The
problem is to find the levels of the ingredients that produce the most
desirable tire tread compound in terms of the four outcome measures.
Design and coding of
variables.
Derringer and Suich (1980) used a central composite design to investigate
the effects of the tire tread ingredients on the desirability of the product.
The data for the completed experiment of 20 runs (as listed on p. 217
of Derringer & Suich, 1980) are contained in the example data file
Tiretre.sta. Open this data file.
Ribbon bar.
Select the Home tab. In
the File group, click the Open arrow and from the menu, select Open Examples to display the Open a Statistica Data File dialog
box. Doubleclick the Datasets
folder, and open the Tiretre.sta data
set.
Classic
menus. On the File menu, select Open Examples to
display the Open a Statistica Data File
dialog box. The Tiretre.sta data
file is located in the Datasets
folder.
In the data file, as shown below, responses on the 4 dependent variables
are in the first 4 columns. Levels of the 3 independent variables are
in the last 3 columns and have already been recoded using the following
formulas:
SILICA = ( phr silica 1.2
) / 0.5
SILANE = ( phr silane  50
) / 10
SULFUR = ( phr sulfur  2.3
) / 0.5
Specifying the design
and fitting the response surface model. The analysis of this data
begins in the same way as for any central composite design (see Example
5: Central Composite (Response Surface) Designs). Start Experimental
Design:
Ribbon bar.
Select
the Statistics
tab, and in the Industrial Statistics group, click
DOE
to display the Design
& Analysis of Experiments Startup
Panel.
Classic
menus. On the Statistics  Industrial Statistics
& Six Sigma submenu, select Experimental
Design (DOE) to display the Design
& Analysis of Experiments Startup Panel.
Doubleclick Central composite, nonfactorial,
surface designs to display the Design
& Analysis of Central Composite (Response Surface) Experiments
dialog box.
Select the Analyze
design tab, and click the Variables
button.
In the variable selection dialog box, select Abrasion,
Modulus, Elong,
and Hardness as the Dependent
variables; select Silica, Silane, and Sulfur
as the Indep. (factors); do not select a Blocking
variable, and click OK.
The current design is a standard, threevariable, rotatable, central
composite design with six center points, and standard coding of the independent
variables is used (so rescaling of the independent variables is not required);
thus, simply click OK to display
the Analysis
of a Central Composite (Response Surface) Experiment dialog
box.
Derringer and Suich (1980) fitted responses on the dependent variables
to a second degree polynomial, response surface model. This is the default
model for standard central composite designs, i.e., the Lin/quad
main eff + 2way interactions option button is selected by default
on the Model
tab.
The fitted coefficients can be produced by selecting each dependent
variable, in turn, from the Variable
dropdown list (located at the top of the dialog box, under the Summary
box), and clicking the Regression coefficients
button on the ANOVA/Effects
tab. The coefficients from the Regression
coefficients spreadsheets
for each of the four dependent variables are shown below (see also p.
218 in Derringer & Suich, 1980).
Note: The
following spreadsheet is not the default way the spreadsheet will be produced
in Statistica. For this image,
four spreadsheets were created
via the Regression coefficients
button (one for each dependent variable), and then the regression coefficient
column from each spreadsheet was copied to a new spreadsheet, shown below.
These are the regression coefficients used in computing predicted values
for the dependent variables at different combinations of levels of the
independent variables.
Now, to profile the predicted responses and the desirability of responses,
select the Prediction
& profiling tab, and click the Response
Desirability Profiling button to display the Profiler.
Response/Desirability Profiling
Specifying desirability
functions. Because we want to profile both the predicted responses
on the dependent variables as well as the overall response desirability,
the first step is to specify the desirability function for each dependent
variable.
To do this, ensure that the Show desirability
function check box is selected, which enables the edit fields for
the Desirability function settings.
Note that the first dependent variable, Abrasion,
is displayed in the Variable
dropdown list, indicating that the desirability function settings you
specify will be applied to predicted values for Abrasion.
Derringer and Suich (1980) specified the desirability function for Abrasion by assigning desirability
values of 0.0 (for undesirable)
to predicted values of Abrasion
below 120, desirability values
of 1.0 (for very desirable) to
predicted values of Abrasion
above 170, and linearly increasing
desirability values between 0.0
and 1.0 for predicted values
of Abrasion between 120
and 170.
This desirability function for Abrasion
is specified by entering the values of 120,
145 (the midpoint value), and
170 in the Low
value, Medium value, and High value edit fields, respectively,
indicating that these predicted values correspond to the three "inflection"
points in the desirability function for Abrasion.
The corresponding default Desirability
values of 0.0, 0.5,
and 1.0 are preset in the Low desirability, Medium
desirability, and High desirability
edit fields, respectively. Because the function does not specify any curvature
in the "fall off" of desirability between inflection points,
the values in the s parameter
and t parameter edit fields can
be left at their default values of 1.0,
specifying linear changes in desirability between inflection points.
The same procedure is followed for specifying the desirability functions
for the remaining three dependent variables: select the dependent variable
for which to supply desirability function specifications in the Variable dropdown list, and then enter
the specifications in the appropriate edit field. Derringer and Suich's
(1980) specifications for the remaining three dependent variables are
shown below.

Low Value 
Med Value 
High Value 
Low Desir 
Med Desir 
High Desir 
s 
t 
Modulus 
1000 
1150 
1300 
0.0 
0.5 
1.0 
1.0 
1.0 
Elong 
400 
500 
600 
0.0 
1.0 
0.0 
1.0 
1.0 
Hardness 
60 
67.5 
75 
0.0 
1.0 
0.0 
1.0 
1.0 
Note that once a set of specifications are entered, they can be saved
by clicking the Save desirability specs
for all variables button on the Save/Open
tab and later retrieved by clicking the Open
desirability specification button, to avoid reentering the same
specifications more than once.
Profiling the predicted
responses and response desirability.
Clicking the View button will
now display a compound response profile graph, using the factor means
as the default current values for each predictor variable, and four steps
from the observed minimum to the observed maximum to define the default
fivegrid points for each factor.
A prediction profile for each dependent variable is shown, consisting
of a series of graphs, one for each independent variable, of the predicted
values for the dependent variable at each grid point of the independent
variable, holding the levels of all other independent variables constant
at their current values. By default, confidence intervals for the predicted
values are also shown.
Graphs of the desirability functions for each dependent variable are
displayed, and a series of graphs, one for each independent variable,
shows the profile of overall response desirability at each grid point
of the independent variable, holding the levels of all other independent
variables constant at their current values. The overall desirability value
of .17956 indicates that the
mean values of the tire tread ingredients do not yield a very desirable
tire tread compound, but the profile of overall response desirability
also shows that a level of Sulfur
lower than its mean could produce a more desirable product.
A search for the levels of the ingredients that produce the most desirable
product can be conducted. This can be specified by selecting the At optimum value option button in the
Set factors at group box.
To conduct the search using a general function optimization method (the
simplex method), select the Options
tab. Select the Use general function
optimization option button in the Search
options (for optimum desirability) group box.
Also, to display finer grids of predicted values and overall response
desirability scores, click the Factor
grid button on the Quick
tab to display the Specifications
for factor grid dialog box. Enter 20
for the No. of steps. Click
the accompanying Apply to all
button to specify a grid of 21 points for each independent variable. Click
OK to return to the Profiler.
Clicking the View button will
now produce a compound response profile graph, using the optimal values
of the independent variables as their current values, and 20 steps from
the observed minimum to the observed maximum to define the 21 grid points
for each factor.
The results displayed on the compound response profile graph show that
desirability is improved by setting the factors at levels other than their
means. The overall desirability value is .5833
with the settings of Silica,
Silane, and Sulfur
at .0515, .1505,
and .8668, respectively. Derringer
and Suich (1980) reported an overall desirability of .583
with the settings of Silica,
Silane, and Sulfur
at .050, .145,
and .868, respectively, using
a FORTRAN search program only briefly described in their article.
As an alternative method for conducting the search, select the Options
tab. In the Search options (for
optimum desirability) group box, select the Optimum
desirability at exact grid points option button. Click the View button to produce a compound response
profile graph, using the optimal values from the grid search as the current
values for the independent variables.
The results displayed on the compound response profile graph show that
the overall desirability value is .57992
with the settings of Silica,
Silane, and Sulfur
at 0.0, .1633,
and .8165, respectively.
With fine grids and large numbers of independent and dependent variables,
the Optimum desirability at exact grid
points option can take a long time to run [in this example 37,044
(or 21 x 21 x 21 x 4) predicted values and corresponding desirability
scores are computed, as well 9,261 overall desirability scores], but it
will always find the optimal settings for the factors.
The Use general function optimization
option is generally faster, but for data sets with several local minima
in overall desirability, or when the maximum desirability is close to
the edge of the experimental region, the Use
general function optimization search option can sometimes fail
to converge or fail to find the maximum desirability (for further details
on Search options, see Special
Topics  Profiling Predicted Responses and Response Desirability and
the Options
tab on the Response/Desirability Profiler).
Surface plots and contour plots for overall response desirability are
available as options in the Profiler.
These plots are useful for interpreting the effects on overall response
desirability of different combinations of levels of each pair of independent
variables, with the remaining independent variables held constant at their
current values.
Clicking the Surface button
(with a number 1, located toward the top of the Profiler)
will produce a compound graph with three 3D surface plots in which pairs
of independent variables are represented on two of the axes and overall
response desirability is represented on the third axis.
Clicking the Contour button
(with a number 2) will produce a compound graph with three contour plots
showing the levels of overall response desirability produced in different
regions of the plane defined by pairs of independent variables, where
each region of the plane represents a different combination of the levels
of the two variables.
All of these plots show that the surface is relatively flat near the
maximum, meaning that small departures from the optimal settings for the
independent variables would not appreciably decrease the desirability
of the product. More generally, these graphic features can aid in distinguishing
between factors that are "inert" and "active" with
respect to other factors. For "inert" ingredients, the desirability
surface or contours are "flat" with respect to an "inert"
ingredient's axis. The desirability surface or contours change at different
levels of an "active" ingredient.
Summary. This
example has shown how to simultaneously optimize several response variables
in a central composite design. The same general procedures are used to
find the levels of the predictor variables that optimize overall response
desirability in other types of designs: 1) fit the observed characteristics
of the product using an appropriate prediction equation based on the levels
of the factors, and 2) find the levels of the factors that simultaneously
produce the most desirable predicted characteristics of the product. The
Experimental
Design module's programs for generating and analyzing designs
help to accomplish the first step, and the Response/Desirability
Profiler is a useful tool for carrying out the important second step
in the process of product development.