Mixture Experiments

Both the Custom Design and Mixture Design platforms construct designs for situations where all of your factors are ingredients in a mixture. However, mixture experiments can involve non-mixture process variables, or process factors. The Custom Design platform can construct a design to accommodate both mixture ingredients and process factors. The Custom Design platform also allows the mixture components to sum to any positive number. See Advanced Options > Mixture Sum.

This section contains the following examples:

•	Mixture Design with Nonmixture Factors

•	Mixture of Mixtures Design

Mixture Design with Nonmixture Factors

In the following example from Atkinson and Donev (1992), you create a design for an experiment involving both mixture factors and process factors. The design is an 18-run design that is balanced with respect to the levels of a categorical factor. The design enables you to fit a full response surface. You use Design Evaluation plots and results to examine the relative prediction variance of the design.

The response and factors involved in the design are as follows:

•	The response is Damping, which measures the electromagnetic damping of an acrylonitrile powder.

•	The three mixture ingredients are:

‒	CuSO4 (copper sulphate), ranging from 0.2 to 0.8

‒	Na2S2O3 (sodium thiosulphate), ranging from 0.2 to 0.8

‒	Glyoxal (glyoxal), ranging from 0 to 0.6

•	The nonmixture environmental factor of interest is Wavelength (the wavelength of light) at three levels denoted L1, L2, and L3.

Wavelength is a continuous variable. However, the researchers were interested only in predictions at three specific wavelengths. For this reason, you treat Wavelength as a categorical factor with three levels.

This example contains the following steps:

•	Create the Design

•	Evaluate the Design

Create the Design

1.	Select DOE > Custom Design.

2.	Double-click Y under Response Name and type Damping.

3.	Click Maximize under Goal and change it to None.

The goal is set to None because the authors of the study do not mention how much damping is desirable.

4.	From the Custom Design red triangle menu, select Load Factors.

5.	Select Help > Sample Data Library and open Design Experiment/Donev Mixture Factors.jmp.

This loads the three mixture ingredients and the categorical process factor. Note that the bounds on the values of the three mixture factors are also loaded.

Responses Outline and Factors Outline

6.	Click Interactions > 2nd.

An informational JMP Alert window reminds you that JMP removes the main effect terms for non-mixture factors that interact with all the mixture factors. This means that the main effect of Wavelength is removed, but all two-way interactions of mixture factors with Wavelength are added.

7.	Click OK to dismiss the message.

The effects in the Model outline define a response surface model in the mixture ingredients along with the additive effect of the wavelength. See Scheffé (1958).

Model and Design Generation Outlines

8.	Leave the default number of runs at 18.

The choice of 18 runs allows six runs for each of the three levels of the wavelength factor.

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 858576648, and click OK.

10.	(Optional) From the Custom Design red triangle menu, select Number of Starts, type 10, and click OK.

11.	Click Make Design.

Design Outline Showing 18-Run Design

You can check that there are six runs for each level of Wavelength.

Evaluate the Design

1.	Open the Design Evaluation > Prediction Variance Profile outline.

Prediction Variance Profile for 18-Run Design

Move the slider for Wavelength to verify that the relative prediction variance profiles for the mixture factors do not change across the levels of Wavelength. Move the slider for any one of the mixture factors. The factors for the other two mixture factors adjust to make the mixture ingredients sum to one. Notice that the smallest relative prediction variances occur near the center settings for the mixture factors.

2.	From the Prediction Variance Profile red triangle menu, select Maximize Desirability.

Notice that the maximum relative prediction variance over the design space is 0.8 times the error variance.

3.	Open the Fraction of Design Space Plot outline.

Fraction of Design Space Plot for 18-Run Design

Over the entire design space, the relative prediction variance is below 0.8. The minimum relative prediction variance is about 0.32. As seen in Prediction Variance Profile for 18-Run Design, the minimum occurs near the center settings for the mixture factors.

4.	Open the Design Diagnostics outline.

Design Diagnostics Outline for 18-Run Design

The design is optimal relative to the D-optimality criterion, even though its D-efficiency is very low (3.6%). Because mixture designs are far from orthogonal due to the mixture constraint, they typically have very low D-efficiencies. The Average (relative) Variance of Prediction is 0.410395. This is consistent with the Fraction of Design Space plot in Fraction of Design Space Plot for 18-Run Design.

Mixture of Mixtures Design

In this example, construct a design for a mixture of mixtures situation.

Consider the ingredients that go into a cake. Dry ingredients include flour, sugar, and cocoa. Wet ingredients include milk, melted butter, and eggs. The wet and dry components of the cake are two mixtures that are first mixed separately and then blended together. Dry and Wet Components and Experimental Ranges lists the factors and the ranges over which you vary them as part of your experiment.

Dry and Wet Components and Experimental Ranges
Mixture	Ingredient	Lower Level	Upper Level
Dry	Cocoa	10% (0.1)	20% (0.2)
	Sugar	0% (0)	15% (0.15)
	Flour	20% (0.2)	30% (0.3)
Wet	Butter	10% (0.1)	20% (0.2)
	Milk	25% (0.25)	35% (0.35)
	Eggs	5% (0.05)	20% (0.20)

The dry components (the mixture of flour, sugar, and cocoa) comprise 45% of the combined mixture. The wet components (butter, milk, and eggs) comprise 55%.

The goal of your experiment is to optimize a Taste rating. Taste is rated on a scale of 1 to 10, with 10 representing the best taste.

You construct a 10-run design to fit a main effects model. Because of the constraint on the proportions of dry and wet ingredients, you need to include only five factors in the Model outline to avoid singularity. The choice of which factor not to include is arbitrary.

This example contains the following steps:

•	Create the Design

•	Analyze the Experimental Results

Create the Design

1.	Select DOE > Custom Design.

2.	Double-click Y under Response Name and type Taste.

Note that the default goal is Maximize. Because you want to maximize the Taste rating, do not change the goal.

3.	Click under Lower Limit and type 0.

The least desirable rating is 0.

4.	Click under Upper Limit and type 10.

The most desirable rating is 10.

5.	Leave the area under Importance blank.

Because there is only one response, that response is given Importance 1 by default.

6.	From the Custom Design red triangle menu, select Load Factors.

7.	Open Cake Factors.jmp from the Design Experiment sample data folder.

Completed Responses and Factors Outlines

Note that the factors are all mixture factors. The Values that define the range of settings for the experiment vary from factor to factor.

8.	In the Define Factor Constraints outline, select Specify Linear Constraints.

9.	In the Linear Constraints panel, click Add twice.

10.	Enter the constraints shown in Define Factor Constraints.

For the second constraint setting, be sure to select the greater than or equal to direction.

Define Factor Constraints

The two inequalities ensure that the dry factors sum to exactly 45% of the total, ensuring that the wet factors constitute the remaining 55%.

11.	In the Model outline, select any effect and click Remove Term.

Because of the equality constraint, a model containing all six effects would be singular.

12.	Type 10 next to User Specified.

Your experiment requires baking 10 cakes.

Note: Setting the Random Seed in step 13 and Number of Starts in step 14 reproduces the exact results shown in this example. In constructing a design on your own, these steps are not necessary.

13.	(Optional) From the Custom Design red triangle menu, select Set Random Seed, type 1992991263, and click OK.

14.	(Optional) From the Custom Design red triangle menu, select Number of Starts, type 40, and click OK.

15.	Click Make Design.

A JMP Alert informs you that your factor constraints include an equality constraint. This was what you intended, because the sum of the dry ingredient proportions is constrained to 45%.

16.	Click OK to dismiss the JMP Alert.

17.	Click Make Table.

Mixture of Mixtures Design

The settings for the dry ingredients sum to 45% of the mixture and the settings for the wet ingredients sum to 55% of the mixture. The settings also conform to the upper and lower limits given in the Factors outline.

Analyze the Experimental Results

The Cake Data.jmp sample data table shows the results of the experiment. The design table contains a Model script that opens a Fit Model window showing the five main effects specified in the DOE window’s Model outline. Notice that the main effect of Egg is not included in the Model outline for this design. This script was saved to the data table when it was created by Custom Design.

1.	Open the Cake Data.jmp sample data table, located in the Design Experiment folder.

2.	In the Tables panel of the design table, click the red triangle next to Model and select Run Script.

The main effect due to Egg is not included because it was excluded from the Model outline in the Custom Design window. All five effects are designated as Response Surface and Mixture effects.

3.	Click Run.

A JMP Alert appears, notifying you that the Profiler cannot be shown because of the additional constraint.

4.	Click OK to dismiss the JMP Alert.

The Parameter Estimates report indicates that Sugar, Flour, and Butter are significant at the 0.05 level.

Parameter Estimates Report