Using
DOE++ For Product and Process Optimization
In
last month’s Reliability Basics article, we discussed
reliability allocation. The purpose of reliability allocation is
to meet the system reliability requirement and, in the meantime,
to keep the cost as low as possible. High reliability usually
means high cost. A desirable design is one that can provide
satisfactory reliability with a minimal cost. This is an
optimization problem with two responses: reliability and cost.
In this article, we will illustrate how to use
DOE++ to optimize several responses simultaneously.
Example:
A chemical engineer is interested in determining the operating
conditions that maximize the yield of a process [1].
At the same time, she also wants the viscosity of the product to
be within the range of 62 to 68, with the target value of 65;
and she wants the molecular weight of the product to be less
than 3400, with the target value of 3000.
Two controllable variables
influence process yield: reaction time and reaction temperature.
From previous experimentation, the engineeer has determined that
the ranges of settings that she wants to examine are given in
the table below.
Name 
Unit 
Level (1) 
Level (+1) 
A: Time 
min 
80 
90 
B:
Temperature 
F 
170 
180 
A central composite design [2,
3] with 5 center points and alpha = 1.414 is used to conduct
the experiment. (Central composite design usually is used to
identify a second order model between input variables and output
response.) Once the model has been identified, it can be used
for optimization.
A quadratic function is used to
establish the relationship between A, B and each response. The
model is:
Since we have three responses
(yield, viscosity and weight), the optimal setting of A and B
for one of the responses may not be good for the other two.
Therefore, a compromise should be made. A balanced setting that
can optimize the overall performance should be found. The
desirability function approach is used to come up with a
balanced solution. This solution tries to satisfy the
requirements for each of the responses as much as possible
without compromising on any of the requirements too much.
Under the desirability function
approach, each response is assigned a desirability function d_{i}.
The value of d_{i}
varies between 0 and 1, with 0 representing that the worst
acceptable value and 1 representing the response that is the
target value. The overall desirability function is defined as:
where r_{i}
represents the relative importance of each response. The bigger
the r_{i} value, the more influence the
corresponding response has on the overall desirability function.
Usually, r_{i} is assigned a value from 0.1 to
10.
Depending on the objective for
the response (maximization, minimization or target value), the
definition of d_{i} is different [2,
3].
For maximization:
For minimization:
For target value:
where:

is the
predicted value for the i^{th}
response.
 T is the target value
for a given response.
 L is the acceptable
lower limit for a response.
 U is the acceptable
upper limit for a response.
 ω
is the weight for a response.
The weight, ω, determines how the
desirability value changes for a response. A value of ω that is less than 1
is equivalent to saying that any response value between the
limit and the target is desirable; a value of ω that is greater than
1 is equivalent to saying that it is very important that the
target is met.
The following steps illustrate
how to use
DOE++ for this chemical process example. The data set for
the example is provided in the Multiple Response Optimization
Folio in the Multiple Response Central Composite Design.rdoe
file, located in the Examples\Response Surface Method folder
within the application directory (e.g. C:\Program
Files\ReliaSoft\DOE\Examples\Response Surface Method).
Create the Design
Step 1: Choose Project > Add Standard Folio.
Step 2: On the first
page of the Design Wizard, select to create a Response
Surface Method Design and click Next to proceed to
the next step.
Step 3: On the second
page of the Design Wizard, select
Central Composite Design and click Next to proceed
to the next step.
Step 4: In the third
step of the Design Wizard, use the following settings:
Step 5: Click the
Factor Properties button. In the Factor Properties window
that appears, enter the time and temperature values, as shown
next, then click OK to close the window and return to the
Design Wizard.
Step 6: Click the
Response Properties button. In the Response Properties
window that appears, define the settings, as shown next, then
click OK to close the window and return to the Design
Wizard.
Step 7: Click Finish
to create the new Folio.
NOTE: The run orders are
randomly generated when you create the design, therefore your
Folio may be different from the Folio in this example.
Step 8: Conduct the
experiment according to the run order in the design matrix and
record all the response values. NOTE: For this example, we will
use the data that is already entered in the Multiple Response
Optimization Folio, as shown next.
Analysis and Results
Once the response values have been entered in the Data Sheet,
you can analyze it. The data set is already entered for you in
the example file.
Step 1: Open the
Multiple Response Optimization Folio.
Step 2: Choose Data >
Select Effects and use the following settings:
Step 3: Click the
Apply to All Responses button to apply the selected effects
to all three responses, then click OK to close the
window.
Step 4: Click the
Calculate icon on the Control Panel to calculate and display
the results.
The Analysis tab provides the
ANOVA table and the Regression Information for each response.
You can switch the displayed results by choosing a different
response on the Control Panel, as shown next.
Step 5: Choose Plot >
Contour/Surface Plot to look at the surface plot to view
where the optimal solution is. For example, for the Yield, the
surface plot is:
From the surface plot, the
approximate location for the mountain peak is at Time = 87 and
Temperature = 177.
In fact, DOE++ has an
optimization tool to help you find the global optimal solution.
The steps for using this tool are presented next.
Optimization
Step 1: Choose Data > Optimization.
Step 2: On the Response
Settings page of the Optimization Settings window, use the
following settings:
Step 3: On the Factor
Settings page of the Optimization Settings window, set the range
of each factor. These default ranges are the limits of each
factor that you used in the experiment.
Step 4: Click OK
to get the optimal solution.
The plot shows where the
optimal solution is and what the desirability values are. This
solution also can be viewed by clicking the
View All Solutions icon.
The Optimization Solutions window will
appear, as shown next.
The optimal settings are Time =
83.92 minutes and Temperature = 170.42F. At this setting, the
predicted yield is 78.3951 with a desirability value of 0.8395;
predicted weight is about 3200 with a desirability value of 0.1;
predicted viscosity is 65 and the desirability is 1. The
achieved overall desirability value is 0.9434.
A followup experiment using
these settings should be conducted to determine whether the
predicted values are accurate. If the response values from the
followup experiment are significantly different from the
predicted values, you need to check whether the followup
experiment was set up correctly. Otherwise, more experiments
should be added in order to determine the true relationship
between factors and responses.
Conclusions
In this article, we illustrated how to use DOE++ to
optimize a chemical process. We also briefly explained the
theory of multiple response optimization and we used the
quadratic model for each response. It is worth mentioning that
this model is validated in the low and high range of each
factor. If the prediction is beyond the range, caution should be
taken because the predicted results may no longer be accurate.
References:
[1] Montgomery, D. C. Design and Analysis of
Experiments, 5th edition, New York: John Wiley & Sons, 2001.
[2] Myers, R. H. Response Surface
Methodology: Process and Product Optimization Using Designed
Experiments, 2nd edition, New York: John Wiley & Sons, 2002.
[3] ReliaSoft Corporation, Design of
Experiment and Analysis Reference, Tucson, AZ: ReliaSoft
Publishing, 2008.
