Response Surface Design With Constraints and Categorical Factor
Use the Custom Design platform to create a design with three factors where there are constraints on factor level combinations. The goal is to optimize the response. The following three factors are to be included in the design:
• Time: The range of interest is 500 to 560 seconds.
• Temperature: The range of interest is 350 to 750 degrees Kelvin.
• Catalyst: Three catalysts A, B, and C, must be tested.
The design must allow for constraints on two of the levels of Catalyst:
• When catalyst B is used, the temperature must be above 400.
• When catalyst C is used, the temperature must be below 650.
Define the Response and Factors
1. Select DOE > Custom Design.
2. In the Response section, double-click Y and change it to Yield.
Because your goal is to maximize Yield, leave the Goal set to Maximize.
3. In the Factors section, type 2 next to Add N Factors.
4. Click Add Factor > Continuous.
5. Click Add Factor > Categorical > 3 Level.
6. Rename the factors Time, Temperature, and Catalyst.
7. Change the Values for Time to 500 and 560.
8. Change the Values for Temperature to 350 and 750.
9. Change the Values for Catalyst to A, B, and C.
Figure 5.42 Factor Settings
10. Click Continue.
Define the Constraints
1. Select Use Disallowed Combinations Filter in the Define Factor Constraints section.
2. Select Temperature and Catalyst from the Add Filter Factors list.
3. Click Add (plus button).
4. Click the 750 that appears below the Temperature slider and change it to 400. Press Enter (or click elsewhere).
This disallows factor settings with Temperature values below 400.
5. Click the B block under Catalyst.
Together with the constraint on Temperature, this disallows factor settings for which Catalyst is B and Temperature is below 400.
6. Click OR.
This enables you to define your second constraint.
7. Select Temperature and Catalyst from the Add Filter Factors list. (Your earlier selection might have been retained.)
8. Click Add (plus button).
9. In the panel that appears beneath the word OR, click the 350 that appears below the Temperature slider and change it to 650. Press Enter (or click elsewhere).
This disallows factor settings with Temperature values above 650.
10. Click the C block under Catalyst.
Together with the constraint on Temperature, this disallows factor settings for which Catalyst is C and Temperature is above 650.
Figure 5.43 Constraints Defined
Note: The (3) afterCatalyst indicates the number of catalyst levels.
Add Response Surface Terms and Make Design
1. In the Model section, select RSM.
JMP adds the appropriate response surface terms to the model.
Note: Setting the Random Seed in step 2 and Number of Starts in step 3 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
2. (Optional) Click the Custom Design red triangle, select Set Random Seed, type 654321, and click OK.
3. (Optional) Click the Custom Design red triangle, select Number of Starts and set it to 1000. Click OK.
4. Click Make Design.
Figure 5.44 Design Satisfying Constraints
Because you added RSM terms to the Model section, this is an I-optimal design. It satisfies your constraints:
– When catalyst B is used, the temperature must be above 400.
– When catalyst C is used, the temperature must be below 650.
View the Design
Recall that the design region consists of settings of Time between 500 and 560 and Temperature between 350 and 750. Generate a graph to obtain a geometric view of the design points within the design region:
1. Click Make Table.
2. Select Graph > Graph Builder.
3. Select Time and Temperature and drag them to the center of the template.
4. Deselect the Smoother icon. This is the second icon from the left above the template.
5. Select Catalyst and drag it to the Group X zone at the top of the template.
Figure 5.45 Design Points for Three Levels of Catalyst
6. Double-click the Temperature axis.
7. Under Reference Lines, add two reference lines:
– Next to Value, enter 350.
– Click Add.
– Next to Value, enter 750.
– Click Add.
8. Click OK.
Figure 5.46 Design Bounds on Temperature for Three Levels of Catalyst
9. Double-click the Time axis.
10. Under Reference Lines, add two reference lines:
– Next to Value, enter 500.
– Click Add.
– Next to Value, enter 560.
– Click Add.
11. Click OK.
12. Click Done.
Note: The plot in Figure 5.47 is vertically resized.
Figure 5.47 Design Regions for Three Levels of Catalyst
Six of the settings for Catalyst A fall on the edges of the design region. All Temperature setting for Catalyst B are above the 400-degree constraint and all Temperature settings for Catalyst C are below the 650-degree constraint.