Example of selecting a D-optimal general full factorial design
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You work for a parts manufacturer and must attach a pipe vertically onto a thin square plate. The heat from welding causes the plate to warp. You want to design an experiment to try to reduce the amount of warping, but you are restricted to an experiment using only 32 runs. Four variables - temperature of the plate (50°, 75°, 100°, 125°), copper content of the plate (40%, 60%, 80%, 100%), size of the end cap on the pipe (0.25", 0.5"), and attachment method (friction weld, soldering) - explain much of the variability in warping.

The full general factorial design for this experiment requires 64 runs. You want to obtain an optimal design that reduces the number of design points to 32.

1    Open the worksheet FACTOPT2.MTW.

2    Choose Stat > DOE > Factorial > Select Optimal Design.

3    In Number of points in optimal design, type 32.

4    Click Terms.

5    From Include terms in the model up through order, choose 2. Click OK in each dialog box.

Session window output

Optimal Design: Temperature, Copper, Endcap, Method

 

 

Factorial design selected according to D-optimality

 

Number of candidate design points: 64

Number of design points in optimal design: 32

 

Model terms: A, B, C, D, 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: 18, 61, 1, 24, 30, 42, 6, 56, 15, 44, 7, 58, 64, 41,

                                      27, 39, 25, 32, 51, 13, 53, 3, 59, 34, 8, 40, 17, 22,

                                      5, 2, 46, 49

 

Condition number:                              223.585

D-optimality (determinant of XTX):         6.43729E+28

A-optimality (trace of inv(XTX)):              11.4062

G-optimality (avg leverage/max leverage):      0.96875

V-optimality (average leverage):               0.96875

Maximum leverage:                                    1

Interpreting the results

The Session window output contains the following five parts:

·    A summary of the D-optimal design. This design was obtained by selecting a subset of 32 points from a candidate set of 64 points.

·    The model terms. D-optimal designs are dependent on the specified model. In this example, the terms include:

A, B, C, D, AB, AC, AD, BC, BD, CD

These are the terms selected in the Terms subdialog box. Remember, a design that is D-optimal for one model will most likely not be D-optimal for another model.

·    The method by which the initial design was generated and whether or not an improvement of the initial design was requested. In this example, the initial design was generated sequentially and the exchange method (using one design point) was used to improve the initial design.

·    The selected design points in the order they were chosen. The numbers shown identify the row of the design points 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 may select a different optimal design from the same set of candidate points if they are in a different order. This can occur because there may be more than one D-optimal design for a given candidate set of points.

·    Minitab displays some variance-minimizing optimality measures. You can use this information to compare designs.