# Mixture DOE

## Summary

Provides a cost-effective methodology for the evaluation of factors whose sum total volume or quantity cannot change. For example, if you wish to add more fruit filling to an 8-ounce fruit bar, another ingredient must be reduced. Such adjustments are common in packaged food and chemical formulations. The goals of this type of experiment are usually focused on developing a full predictive model (Y = f(X)) describing how the ingredients in the mixture jointly affect the process output and determining the optimal amounts of each ingredient.

• Which ingredients have the largest effects on the process output (which ingredients are the key ingredients)?
• Do important interactions exist between ingredients?
• Which process inputs have the largest effects on the process output (which inputs are the key inputs)?
• Do important interactions exist between process inputs?
• How much variation in the process output can be explained by varying the ingredients and process inputs?
• What is the optimal combination of ingredients?
• What are the optimal settings of the process inputs?
When to Use Purpose
Mid-project If you believe the desired characteristics of the mixture are a function of only the ingredients, use a pure mixture DOE to evaluate which ingredients have the largest influence on the characteristics, build a predictive model using the key ingredients, and find the optimal quantities of the ingredients.
Mid-project If you believe the desired characteristics of the mixture are a function of both the ingredients and the process, use a mixed model DOE (some factors are ingredients, some are process inputs) to evaluate which ingredients and process inputs have the largest influence on the characteristics. Then, build a predictive model using the key ingredients and key process inputs and find the optimal quantities of the ingredients along with the optimal settings of the process inputs.

### Data

Continuous Y, continuous X's (for a pure mixture design).

## How-To

1. State your factors (ingredients) and any constraints they may have (for example, fruit bars must have at least 30% peanuts but not more than 50%).
2. Verify the measurement systems for the Y data and the ingredients are adequate.
3. Develop a data collection strategy (who should collect the data, as well as where and when; how many data values are needed; the preciseness of the data; how to record the data, and so on).
4. Run the experiment and reduce to a final model by eliminating interaction terms (such as binary blending terms) with high p-values (typically greater than 0.05). All linear terms must stay in the model, because they are part of the formulation; removing a linear term would mean you remove an ingredient from the mixture.
5. Use either the response optimizer or mixture contour plots to determine optimal settings of the ingredients.
6. Generate the prediction equation.

## Guidelines

• First, you should develop a sound data collection strategy to ensure you are basing your conclusions on truly representative data.
• Whenever possible, you should do the runs in the experiment in random order to prevent confusing a factor effect with the effect of an untested factor (sometimes called a lurking variable).
• The residuals of the final model must be reasonably normal and with reasonably equal variance. The residuals are usually analyzed by a histogram, normal probability plot, residuals versus fits, and residuals versus order, which can be run at one time using the Four in one option.
• Minitab allows the use of mixed model designs (mixture-process experiments) in which you use a combination of traditional and mixture DOE approaches. For example, a 1-pound cake recipe has six ingredients as part of its mixture component and has the process variables temperature and time as part of its standard DOE. Note: The process variables (X's) can be discrete (such as fan on or off).
• Minitab also allows a mixture DOE analysis in which the relative proportions of the components as well as the total volume of the mixture are analyzed in the same design (mixture-amounts experiments). For example, use the cake example from above, evaluate the results when you bake 1-pound, 2-pound, and 3-pound cakes.
• Check for possible outliers in the unusual observations table (Session window output).
• Do not extrapolate beyond your inference space.
• While the discussion here focuses on Minitab designed experiments, note that you can use the “analyze” portion of the mixture DOE to analyze any numeric experimental data (for example, two factors each at 10 levels). To do this, enter the Y and X data into Minitab, and then use Stat > DOE > Mixture > Analyze Mixture Design to define the factors. You can then analyze this newly defined, custom design in the usual manner.
• If you have discrete numeric data from which you can obtain every equally spaced value and you have measured at least 10 possible values, you can evaluate these data as if they are continuous.

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