# Methods and formulas for Select Optimal Design

Select the method or formula of your choice.

## D-optimality

The D-optimality criterion minimizes the determinant of the variance-covariance matrix of the regression coefficients. You specify the model, then Minitab selects design points that optimize the D-optimal criterion from a set of candidate points. The rows of design columns in the worksheet contain the candidate set of design points. The selection process consists of two steps:

• Generating an initial design
• Improving the initial design to obtain the final design

### Selecting an optimal design

Minitab selects design points from the candidate set to obtain the initial design. You can choose which algorithm will be used to select these points in the Methods sub-dialog box. Choices include: sequential selection, random selection, or a combination of sequential and random selection. By default, Minitab selects all points sequentially. Sequential selection means that all of the points in the initial design were added in an order that provided the maximum increase in D-optimality. If you repeat the design selection and the runs that are in the candidate set are in the same order, the algorithm will find the same solution.

Minitab then tries to improve the initial design by adding and removing points to obtain the final design (referred to simply as the optimal design). You can choose the improvement method in the Methods sub-dialog box. Choices include:
• Exchange method. Minitab first adds the best points from the candidate set, and then drops the worst points until the D-optimality of the design cannot be improved further. You can specify the number of points to be exchanged in the Methods sub-dialog box.
• Fedorov's method. Minitab will simultaneously switch pairs of points. This is accomplished by adding one point from the candidate set and dropping another point so that the switch results in maximum improvement in D-optimality. This process continues until the design cannot be improved further.
• None. In this case, the final design is the same as the initial design.
By default, Minitab improves the design by exchanging one point at a time.

### Augmenting an optimal design

The initial design can be obtained in one of two ways:
• You can use all the design points in the worksheet for the initial design.
• You can use an indicator column to specify which design points and how many replicates of each point comprise the initial design.
If you are augmenting the design, Minitab adds the "best" points in the candidate set sequentially.
Minitab then tries to improve the initial design by adding and removing points to obtain the final design (referred to simply as the optimal design). You can choose the improvement method in the Methods sub-dialog box. Choices include:
• Exchange method. Minitab will first add the best points from the candidate set, and then drop the worst points until the D-optimality of the design cannot be improved further. You can specify the number of points to be exchanged in the Methods sub-dialog box.
• Fedorov's method. Minitab will simultaneously switch pairs of points. This is accomplished by adding one point from the candidate set and dropping another point so that the switch results in maximum improvement in D-optimality. This process continues until the design cannot be improved further.
• Suppress improvement of the initial design. In this case, the final design will be the same as the initial design.
By default, Minitab improves the design by exchanging one point.

Candidate design points may be added with replacement to the final design during the optimization procedure. Therefore, the final design may contain duplicate design points.

###### Tip

In numerical optimization, there is always a danger of finding a local optimum instead of the global optimum. To avoid finding a local optimum, you could perform multiple trials of the optimization procedure starting from different initial designs. Only one trial is possible if you generate the initial design by purely sequential selection or if you specify the initial design with an indicator column.

Minitab identifies the design with the highest D-optimality, and performs the following steps for this design:
• Creates an indicator column (OptPoint) in the original worksheet that shows whether or not a point was selected and the number of replicates of that design point.
• Copies the selected design points to a new worksheet.

## Distance-based optimality

If you do not want to select a model in advance, a good strategy is to spread the design points uniformly over the design space. In this case, the distance-based method provides one solution for selecting the design points. The distance-based optimality algorithm selects design points from a candidate set, such that the points are spread evenly over the design space.

The algorithm for distance based designs does not use an exchange method. The algorithm also does not replicate points when you select an optimal design.

### Selecting an optimal design

Minitab selects the candidate point with the largest Euclidean distance from the origin (response surface design) or the point that is closest to a pure component (mixture design) as the starting point. Then, Minitab adds additional design points in a stepwise manner such that each new point is as far as possible from the points already selected for the design.

### Augmenting an optimal design

You must use an indicator column to indicate which points are available to add to the original design. Then, Minitab adds additional design points in a stepwise manner such that each new point is as far as possible from the points already selected for the design.

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