You usually want to add points between the vertices. The average of points that share a constraint boundary is called a centroid point, and centroid points of various degrees can be added. The centroid point for two neighboring vertices joined by a line is a second degree centroid because a line is two dimensional. The centroid point for vertices sharing a plane is a third degree centroid because a plane is three dimensional, and so on.
1.
Select DOE > Mixture Design.
2.
Enter factors and responses. These steps are outlined in Enter Responses and Factors into the Custom Designer. If your factor ranges are constrained, enter the limits as upper and lower limits in the Factors panel (see Example of Five-factor Extreme Vertices).
3.
Click Continue.
4.
If you have linear constraints, click Linear Constraints and enter them.
5.
In the Degree text box, enter the degree of the centroid point you want to add. The centroid point is the average of points that share a constraint boundary.
6.
If you have linear constraints, click the Linear Constraints button for each constraint you want to add. Use the text boxes that appear to define a linear combination of factors to be greater or smaller than some constant.
7.
Click Extreme Vertices to see the factor settings.
Keep the Same—the rows (runs) in the output table will appear as they do in the Design panel.
Sort Left to Right—the rows (runs) in the output table will appear sorted from left to right.
Randomize—the rows (runs) in the output table will appear in a random order.
Sort Right to Left—the rows (runs) in the output table will appear sorted from right to left.
Randomize within Blocks—the rows (runs) in the output table will appear in random order within the blocks you set up.
9.
Enter the sample size you want in the Choose desired sample size text box.
10.
(Optional) Click Find Subset to generate the optimal subset having the number of runs specified in sample size box described in Step 8. The Find Subset option uses the row exchange method (not coordinate exchange) to find the optimal subset of rows.
11.
Click Make Table.
1.
Select DOE > Mixture Design.
3.
Click Continue.
Example of Five-factor Extreme Vertices
5.
Click Extreme Vertices.
6.
Select Sort Left to Right from the Run Order menu.
7.
Click Make Table.
JMP Design Table for Extreme Vertices with Range Constraints shows a partial listing of a resulting design. Note that the Rows panel in the data table shows that the table has the default 116 runs.
JMP Design Table for Extreme Vertices with Range Constraints
8.
Click Back, then click Continue.
9.
Enter 4 in the Degree text box and click Extreme Vertices.
11.
Click Find Subset to generate an optimal subset having the number of runs specified.
The resulting design (JMP Design Table for 10-Run Subset of the 116 Current Runs) is an optimal 10-run subset of the 116 current runs. This is useful when the extreme vertices design generates a large number of vertices. Your design may look different, because there are different subsets that achieve the same D-efficiency.
JMP Design Table for 10-Run Subset of the 116 Current Runs
Note: The Find Subset option uses the row exchange method (not coordinate exchange) to find the optimal subset of rows.
X1 0.1
X1 0.5
X2 0.1
X2 0.7
X3 0.7
90 85*X1 + 90*X2 + 100*X3
.4 0.7*X1 + X3
2.
Click Continue.
3.
Click the Linear Constraint button three times. Enter the constraints as shown in Constraints.
4.
Click the Extreme Vertices button.
5.
Change the run order to Sort Right to Left, and keep the sample size at 13. See Constraints for the default Factor Settings and completed Output Options.
6.
Click Make Table.
Constraints
This example is best understood by viewing the design as a ternary plot, as shown at the end of this chapter, in Diagram of Ternary Plot Showing Piepel Example Constraints. The ternary plot shows how close to one a given component is by how close it is to the vertex of that variable in the triangle. See Creating Ternary Plots, for details.
The XVERT method first creates a full 2nf 1 design using the given low and high values of the nf – 1 factors with smallest range. Then, it computes the value of the one factor left out based on the restriction that the factors’ values must sum to one. It keeps the point if it is in that factor’s range. If not, it increments or decrements it to bring it within range, and decrements or increments each of the other factors in turn by the same amount, keeping the points that still satisfy the initial restrictions.
The above algorithm creates the vertices of the feasible region in the simplex defined by the factor constraints. However, Snee (1975) has shown that it can also be useful to have the centroids of the edges and faces of the feasible region. A generalized n-dimensional face of the feasible region is defined by nf – n of the boundaries and the centroid of a face defined to be the average of the vertices lying on it. The algorithm generates all possible combinations of the boundary conditions and then averages over the vertices generated on the first step.