Construct a response surface design for three continuous factors that you have identified as active. You want to find process settings to maintain your response(Y) within specifications. The lower and upper specification limits for Y are 54 and 56, respectively, with a target of 55.
1.
Select DOE > Custom Design.
2.
In the Responses outline, click Maximize and select Match Target.
3.
Type 54 as the Lower Limit and 56 as the Upper Limit.
4.
Leave Importance blank.
5.
Type 3 next to Add N Factors.
6.
Click Add Factor > Continuous.
7.
Click Continue.
Note: Setting the Random Seed in step 9 and Number of Starts in step 10 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
9.
(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 929281409, and click OK.
10.
(Optional) From the Custom Design red triangle menu, select Number of Starts, type 40, and click OK.
11.
Click Make Design.
RSM Design
12.
Open the Design Evaluation > Design Diagnostics outline.
Design Diagnostics Outline
13.
Open the Design Evaluation > Prediction Variance Profile outline.
Prediction Variance Profile
The vertical axis shows the relative prediction variance of the expected value of the response. The relative prediction variance is the prediction variance divided by the error variance. When the relative prediction variance is one, its absolute variance equals the error variance of the regression model.
14.
Select Maximize Desirability from the red triangle menu next to Prediction Variance Profile outline.
Prediction Variance Profile with Relative Variance Maximized
15.
Open the Design Evaluation > Fraction of Design Space Plot outline.
Fraction of Design Space Plot
The Custom RSM.jmp sample data table contains the results of the experiment. The Model script opens a Fit Model window showing all of the effects specified in the DOE window’s Model outline. This script was saved to the data table by the Custom Design platform.
1.
Select Help > Sample Data Library and open Design Experiment/Custom RSM.jmp.
3.
Click Run.
Effect Summary Report
The report shows that X1, X2, X1*X1, and X2*X2 are significant at the 0.01 level. None of the other effects are significant at even the 0.10 level. Reduce the model by removing these insignificant effects.
4.
In the Effect Summary report, select X3, X1*X2, X3*X3, X1*X3, and X2*X3.
Effect Summary Report with Insignificant Effects Selected
5.
Click Remove.
Use the Prediction Profiler (at the bottom of the Fit Least Squares window) to explore how the predicted response (Y) changes as you vary the factors X1 and X2. Note the quadratic behavior of Y across the values of X1 and X2.
Remember that you entered response limits for Y in the Responses outline of the Custom Design window. As a result, the Response Limit column property is attached to the Y column in the design table. The Desirability function for Y (in the top plot at right) is based on the information contained in the Response Limit column property. JMP uses this function to calculate Desirability as a function of the settings of X1 and X2. The traces of the Desirability function appear in the bottom row of plots.
6.
In the Prediction Profiler report, select Maximize Desirability from the red triangle options.
Prediction Profiler with Desirability Maximized
The predicted response achieves the target value of 55 at the process settings shown in red above X1 and X2. Prediction Profiler with Desirability Maximized shows that a value of X1 near 0.65 also achieves a predicted value of 55 when X2 = -0.75062. In fact, your Prediction Profiler might show different settings as those that maximize desirability. This is because the predicted response is 55 for many settings of X1 and X2.
7.
Select Factor Profiling > Contour Profiler from the red triangle next to Response Y.
Contour Profiler
The settings of X1 and X2 that correspond to the red contour have predicted response values of 55. You might want to select from among these process settings based on cost efficiency.
1.
Select DOE > Custom Design.
2.
Type 2 next to Add N Factors.
3.
Click Add Factor > Continuous.
4.
Click Add Factor > Blocking > 4 runs per block.
Factors Outline with Two Continuous Factors and a Blocking Factor
5.
Click Continue.
The Default number of runs is 12. The Factors outline updates to show three levels for the Blocking factor, X3. Because you required X3 to have four runs per block, the 12 runs allow three blocks.
6.
Click RSM.
7.
Click OK to dismiss the message.
Quadratic and interactions terms for X1 and X2 are added to the model. Because you added RSM terms, the Recommended optimality criterion changes from D-Optimal to I-Optimal. You can see this later in the Design Diagnostics outline.
Model Outline with Response Surface Effects
Note: Setting the Random Seed in step 8 and Number of Starts in step 9 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
8.
(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 1415408414, and click OK.
9.
(Optional) From the Custom Design red triangle menu, select Number of Starts, type 21, and click OK.
10.
Click Make Design.
11.
Open the Design Evaluation > Design Diagnostics outline.
Design Diagnostics Outline
12.
Click Make Table.
Design Table with Blocking Factor
Fit Model Window
The blocking factor (X3) is entered as an effect.
1.
Select DOE > Custom Design.
2.
Type 2 next to Add N Factors.
3.
Click Add Factor > Continuous.
4.
Click Continue.
5.
Click RSM.
Quadratic and interactions terms for X1 and X2 are added to the model. Because you added RSM terms, the Recommended optimality criterion changes from D-Optimal to I-Optimal. You can see this later in the Design Diagnostics outline.
Note: Setting the Random Seed in step 6 and Number of Starts in step 7 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.
6.
(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 383570403, and click OK.
7.
8.
Click Make Design.
I-Optimal Design
9.
Open the Design Evaluation > Prediction Variance Profile outline.
Prediction Variance Profile for I-Optimal Model
10.
Open the Fraction of Design Space Plot outline.
The Fraction of Design Space Plot appears on the left in Fraction of Design Space Plots (I-Optimal on left, D-Optimal on right). When the Fraction of Space is 0.95, the vertical coordinate of the blue curve is about 0.5. This means that for about 95% of the design space, the relative prediction variance is below 50% of the error variance.
2.
4.
From the red triangle next to Custom Design, select Optimality Criterion > Make D-Optimal Design.
5.
Click Make Design.
D-Optimal Design
6.
Open the Design Evaluation > Prediction Variance Profile outline.
Prediction Variance Profile for D-Optimal Model
At the center of the design region, the relative prediction variance is 0.53562, as compared to 0.208333 for the I-optimal design (Prediction Variance Profile for I-Optimal Model). This means that the relative standard error is 0.732 for the D-optimal design and 0.456 for the I-optimal design. All else being equal, at the center of the design region, confidence intervals for the expected response based on the D-optimal design are about 60% wider than those based on the I-optimal design.
The Design outline shows that the D-optimal design has nine design points, one for every combination of X1 and X2 set to -1, 0, 1. The D-optimality criterion attempts to keep the relative prediction variance low at each of these design points. Explore the variance at the extremes of the design region by moving the sliders for X1 and X2 to -1 and 1. Note that the variance at these extreme points is usually smaller than the variance for the I-optimal design at these points.
7.
Open the Design Evaluation > Fraction of Design Space Plot outline.
Fraction of Design Space Plots (I-Optimal on left, D-Optimal on right)
Fraction of Design Space Plots Superimposed