5.
Process Improvement
5.5.
Advanced topics
5.5.2.
What is a computer-aided design?
5.5.2.1.
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D-Optimal designs
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D-optimal designs are often used when classical
designs do not apply or work
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D-optimal designs are one form of design provided by a computer
algorithm. These types of computer-aided designs are particularly
useful when classical designs do not apply.
Unlike standard classical designs such as factorials and fractional
factorials, D-optimal design matrices are usually not orthogonal and
effect estimates are correlated.
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These designs are always an option regardless of model or resolution
desired
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These types of designs are always an option regardless of the type of
model the experimenter wishes to fit (for example, first order,
first order plus some interactions, full quadratic, cubic, etc.) or
the objective specified for the experiment (for example, screening,
response surface, etc.). D-optimal designs are straight optimizations
based on a chosen optimality criterion and the model that will be fit.
The optimality criterion used in generating D-optimal designs is one of
maximizing |X'X|, the determinant of the information matrix X'X.
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You start with a candidate set of runs and the algorithm chooses
a D-optimal set of design runs
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This optimality criterion results in minimizing the generalized variance
of the parameter estimates for a pre-specified model. As a result, the
'optimality' of a given D-optimal design is model dependent. That is,
the experimenter must specify a model for the design before a computer
can generate the specific treatment combinations. Given the total
number of treatment runs for an experiment and a specified model, the
computer algorithm chooses the optimal set of design runs from a
candidate set of possible design treatment runs. This candidate
set of treatment runs usually consists of all possible combinations of
various factor levels that one wishes to use in the experiment.
In other words, the candidate set is a collection of treatment
combinations from which the D-optimal algorithm chooses the treatment
combinations to include in the design. The computer algorithm generally
uses a stepping and exchanging process to select the set of treatment
runs.
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No guarantee
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Note: There is no guarantee that the design the computer
generates is actually D-optimal.
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D-optimal designs are particularly useful when resources are
limited or there are constraints on factor settings
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The reasons for using D-optimal designs instead of standard classical
designs generally fall into two categories:
- standard factorial or fractional factorial designs require too
many runs for the amount of resources or time allowed for the
experiment
- the design space is constrained (the process space contains
factor settings that are not feasible or are impossible to
run).
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Industrial example demostrated with JMP software
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Industrial examples of these two situations are given below and the
process flow of how to generate and analyze these types of designs is
also given. The software package used to demonstrate this is JMP
version 3.2. The flow presented below in generating the design
is the flow that is specified in the JMP Help screens under its
D-optimal platform.
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Example of D-optimal design: problem setup
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Suppose there are 3 design variables (k = 3) and engineering
judgment specifies the following model as appropriate for the process
under investigation
The levels being considered by the researcher are (coded)
X1: 5 levels (-1, -0.5, 0, 0.5, 1)
X2: 2 levels (-1, 1)
X3: 2 levels (-1, 1)
One design objective, due to resource limitations, is to use
n = 12 design points.
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Create the candidate set
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Given the above experimental specifications, the first thing to do
toward generating the design is to create the candidate set. The
candidate set is a data table with a row for each point (run) you want
considered for your design. This is often a full factorial. You can
create a candidate set in JMP by using the Full Factorial design given
by the Design Experiment command in the Tables menu. The candidate set
for this example is shown below. Since the candidate set is a full
factorial in all factors, the candidate set contains (5)*(2)*(2) = 20
possible design runs.
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Table containing the candidate set
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TABLE 5.1 Candidate Set for Variables X1, X2, X3
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X1
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X2
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X3
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-1
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-1
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-1
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-1
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-1
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+1
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-1
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+1
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-1
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-1
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+1
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+1
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-0.5
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-1
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-1
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-0.5
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-1
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+1
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-0.5
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+1
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-1
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-0.5
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+1
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+1
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0
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-1
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-1
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0
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-1
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+1
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0
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+1
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-1
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0
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+1
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+1
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0.5
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-1
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-1
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0.5
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-1
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+1
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0.5
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+1
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-1
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0.5
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+1
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+1
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+1
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-1
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-1
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+1
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-1
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+1
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+1
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+1
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-1
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+1
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+1
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+1
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Specify (and run) the model in the Fit Model dialog
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Once the candidate set has been created, specify the model you want
in the Fit Model dialog. Do not give a response term for the model!
Select D-Optimal as the fitting personality in the pop-up menu at the
bottom of the dialog. Click Run Model and use the control panel that
appears. Enter the number of runs you want in your design (N=12 in this
example). You can also edit other options available in the control
panel. This control panel and the editable options are shown in the
table below. These other options refer to the number of points chosen
at random at the start of an excursion or trip (N Random), the number of
worst points at each K-exchange step or iteration (K-value), and the
number of times to repeat the search (Trips). Click Go.
For this example, the table below shows how these options were set
and the reported efficiency values are relative to the best
design found.
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Table showing JMP D-optimal control panel and efficiency report
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D-Optimal Control Panel
Optimal Design Controls
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N Desired
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12
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N Random
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3
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K Value
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2
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Trips
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3
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Best Design
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D-efficiency
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68.2558
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A-efficiency
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45.4545
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G-efficiency
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100
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AvgPredSE
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0.6233
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N
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12.0000
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The algorithm computes efficiency numbers to zero in on a D-optimal
design
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The four line efficiency report given after each search shows the best
design over all the excursions (trips). D-efficiency is the objective,
which is a volume criterion on the generalized variance of the
estimates. The efficiency of the standard fractional factorial is
100%, but this is not possible when pure quadratic terms such as
(X1)2 are included in the model.
The efficiency values are a function of the number of points in the
design, the number of independent variables in the model, and the
maximum standard error for prediction over the design points. The best
design is the one with the highest D-efficiency. The A-efficiencies
and G-efficiencies help choose an optimal design when multiple
excursions produce alternatives with similar D-efficiency.
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Using several excursions (or trips) recommended
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The search for a D-optimal design should be made using several
excursions or trips. In each trip, JMP 3.2 chooses a different set
of random seed points, which can possibly lead to different designs.
The Save button saves the best design found. The standard error of
prediction is also saved under the variable OptStdPred in the table.
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The selected design should be randomized
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The D-optimal design using 12 runs that JMP 3.2 created is listed below
in standard order. The design runs should be randomized before the
treatment combinations are executed.
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Table showing the D-optimal design selected by the JMP software
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TABLE 5.2 Final D-optimal Design
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X1
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X2
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X3
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OptStdPred
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-1
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-1
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-1
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0.645497
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-1
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-1
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+1
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0.645497
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-1
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+1
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-1
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0.645497
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-1
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+1
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+1
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0.645497
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0
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-1
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-1
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0.645497
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0
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-1
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+1
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0.645497
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0
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+1
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-1
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0.645497
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0
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+1
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+1
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0.645497
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+1
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-1
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-1
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0.645497
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+1
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-1
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+1
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0.645497
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+1
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+1
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-1
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0.645497
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+1
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+1
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+1
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0.645497
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| Parameter
estimates are usually correlated |
To see the correlations of the parameter estimates for the best design
found, you can click on the Correlations button in the D-optimal
Search Control Panel. In most D-optimal designs, the correlations
among the estimates are non-zero. However, in this particular example,
the correlations are zero.
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Other software may generate a different D-optimal design
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Note: Other software packages (or even other releases of JMP)
may have different procedures for generating D-optimal designs - the
above example is a highly software dependent illustration of how to
generate a D-optimal design.
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