5.5.2.1. D-Optimal designs

5. Process Improvement
5.5. Advanced topics
5.5.2. What is a computer-aided design?

5.5.2.1. D-Optimal designs

D-optimal designs are often used when classical designs do not apply or work

These designs are always an option regardless of model or resolution desired

You start with a candidate set of runs and the algorithm chooses a D-optimal set of design runs

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.

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 criteria 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.

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 for the design. 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 actual treatment combinations to include in the design. The computer algorithm generally uses a stepping and exchanging process to determine which treatment runs the design will consist of.

Note: There is no guarantee that the resulting design the computer generates is actually optimal.

D-optimal designs are particularly useful when resources are limited or there are constraints on factor settings

The reasons for using D-optimal designs instead of standard classical designs generally fall into one of two categories:

standard factorial or fractional factorial designs require too many runsfor 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).

Industry examples of these two situations are given below and the process flow of how to generateand analyze these types of designs is 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.

Examples of D-optimal designs

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)

One design objective, due to resource limitations, is to use n = 12 design points.

Given the above experimental specifications, the first thing to do towards generating the design isto 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.

TABLE 5.1 Candidate Set for Variables X1, X2, X3

X1	X2	X3	response

-1	-1	-1	?
-1	-1	1	?
-1	1	-1	?
-1	1	1	?
-0.5	-1	-1	?
-0.5	-1	1	?
-0.5	1	-1	?
-0.5	1	1	?
0	-1	-1	?
0	-1	1	?
0	1	-1	?
0	1	1	?
0.5	-1	-1	?
0.5	-1	1	?
0.5	1	-1	?
0.5	1	1	?
1	-1	-1	?
1	-1	1	?
1	1	-1	?
1	1	1	?

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 what these options were set at and the reported efficiency values correspond to the best design found so far.

D-Optimal Control Panel
Optimal Design Controls

N Desired          12
N Random           3
K Value                2
Trips                     3

Best Design

D-efficiency       68.2558
A-efficiency       45.4545
G-efficiency    100
AvgPredSE       0.6233
N                       12.0000

The algorithm computes efficiency numbers to zero in on a D-optimal design

The four line efficiency report given after each search shows the best design over all the excursions (trips).D-efficiency is the objective. It 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 X1²are 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.

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.

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.

TABLE 5.2 Final D-optimal Design

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.

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.