Example of selecting a D-optimal response surface design

A materials scientist has determined four factors that explain much of the variability in the rate of crystal growth. The scientist designs a central composite response surface experiment to define the optimal conditions for crystal growth. After creating the design, the scientist determines that available resources restrict the number of design points that can be included to 20.

The original design the scientist had planned was a central composite design for four factors in two blocks. The four factors are:

Time the crystals are exposed to a catalyst
Temperature in the exposure chamber
Pressure within the chamber
Percentage of the catalyst in the air of the chamber

The blocks represent the plan to run the design sequentially, first assessing factorial and center points. Depending on the analysis of the first block, the scientist could choose to run the points in the axial block to add quadratic terms to the model.

The scientist wants to use D-optimality as a criterion for selecting 20 points from the original design that follow the original blocking scheme and allow estimation of the terms the scientist planned to study with the complete central composite design.

Open the sample data, CrystalGrowth_optimal_design.MTW. This worksheet contains the response surface design that was created in Example of central composite design.
Choose Stat > DOE > Response Surface > Select Optimal Design.
In Number of points in optimal design, type 20.
Click Terms.
Click OK in each dialog box.

Interpret the results

The output contains several components, as follows:

Summary of the D-optimal design

This design is a subset of 20 experimental runs from a candidate set of 30 experimental runs.

Model terms

D-optimal designs depend on the specified model. In these results, the terms include the full quadratic terms that are default in the Terms sub-dialog box. The terms are as follows:

Block A B C D AA BB CC DD AB AC AD BC BD CD

Remember, a design that is D-optimal for one set of terms is not necessarily D-optimal for a different set of terms.

Methods to select the design

Minitab selects the optimal design in two phases.

The first phase selects an initial design with the correct number of runs, either using sequential optimization or including some randomly selected runs.
The second phase improves on this design using the exchange method or Federov's method.

In this example, the initial design is generated by sequential optimization. The optimality is improved using the exchange method. In the exchange method, one point is exchanged at each step.

Experimental runs in the order that they were chosen

The numbers shown identify the row of the experimental run in the original worksheet.

Note

The design points that are selected depend on the row order of the points in the candidate set. Therefore, Minitab can select a different optimal design from the same set of candidate points if they are in a different order. This can occur because multiple D-optimal designs can exist for a specified candidate set of points.

Statistics

You can use optimality metrics to compare designs, but remember that the optimality of a given D-optimal design is model dependent. That is, optimality is defined for a fixed design size and for a particular model. For instance, when comparing designs, larger D-optimality are better, but smaller A-optimality values better.

Response surface design selected according to D-optimality

Number of candidate design points: 30

Number of design points in optimal design: 20

Model terms: Block, A, B, C, D, AA, BB, CC, DD, AB, AC, AD, BC, BD, CD

Initial design generated by Sequential method

Initial design improved by Exchange method

Number of design points exchanged is 1

Optimal Design

Row number of selected design points: 22, 23, 25, 27, 4, 8, 19, 2, 14, 15, 13, 6, 9, 3, 16,
24, 28, 30, 26, 1

Condition number:	10.2292
D-optimality (determinant of XTX):	2.73819E+18
A-optimality (trace of inv(XTX)):	2.50391
G-optimality (avg leverage/max leverage):	0.8
V-optimality (average leverage):	0.8
Maximum leverage:	1