0. Three-Dimensional Variance Dispersion Graphs for Mixture-Process Experiments with Control and Noise Variables
Heidi B. Goldfarb – The Dial Corporation
Douglas C. Montgomery – Arizona State University
Connie Borror – Arizona State University
Christine Anderson-Cook – Virginia Tech

1. Outline Background Model Development Variance Dispersion Graphs
Three-Dimensional Variance Dispersion Graphs
Examples
Conclusions

2. Mixture Designs
Mixture designs are used when the experimental variables have additional constraints on them and it is the proportions of the variables that is important, not the absolute amounts
For example, consider a 6 oz. fish patty that is a comprised of three different types of fish
The proportions of the fish types affect the texture of the patty
The goal is to find the proportions of the three fish types that makes a patty with the firmest texture
Cornell (2002) gives a comprehensive treatment of mixture designs

3. Robust Designs
Robust designs are used when we have noise variables - variables that are uncontrollable, difficult to control, or out of our control in practice
Example: The fish patties are sold to people who cook them. Although recommended temperatures and times are given, we know that not all people follow them exactly
The goal is to find patties that will have a firm texture throughout a range of temperature / time conditions
Concepts of robust design were introduced in the United States in the 1980’s by Taguchi

4. MPV Experiments with Control and Noise Variables
We could also have controllable processing variables, such as the amount of time the patties are precooked before being packaged and sold
Standard designs generate blends with simplex or D-optimal designs and look at each blend at all possible combinations of the processing and noise variables (or a carefully chosen subset thereof)
Steiner and Hamada (1997) address this problem but with different models and without correlation among the noise variables
Goldfarb, Montgomery, and Borror (2003) uses the model shown here and considers correlation

5. Basic Problem
What blend of fish and pre-cooking time will be “best” under a wide range of temperature and time?
What type of experimental designs can help find this combination?
Which designs will build prediction models with the smallest variation?

6. Mixture-Process-Noise Model
Mixture Components xi i = 1, 2, …, q
Controllable Process Variables wp p = 1, 2, …, c
Noise Variables zt t = 1, 2, …, n
where V is a cn x n block diagonal matrix with w’s on the diagonals and 0’s elsewhere

7. Expected Value and Variance
Using the Delta Method:

8. Prediction Variance
For a given design, the standardized prediction variance at a given point, x0, is:
The scaled version (SPV) allows for fair comparisons among designs with different numbers of runs:

9. Mean and Slope Variance Models
Borror, Montgomery, and Myers (2002) develop mean and slope prediction models for RSM designs with noise variables
The mean model variance assesses the prediction error variance taking in to account both model errors and the variation transmitted through the noise variables
The slope model variance measures the precision of the variance of the full model

10. Prediction Variance for MPV Designs with Control and Noise Variables
Recall the model:
The prediction variance for the mean model is:
where C is inverse (X*´X*) matrix and X* is the full model form matrix with x, w, and z terms

11. C-Matrix

12. Mean Model
We divide by σ2 and multiply by N to allow for comparisons of designs, including those with different numbers of runs
k2a and k2b represent the interaction between the mixture-noise and mixture-control-noise variables, respectively
it= k2a and ipt= k2b.

13. Slope Model
For the slope model we need to look at the partial derivative of the model with respect to each of the noise variables. For our model this derivative is:
and

14. General Form for the Slope
For a quadratic mixture model with linear control and noise variables, the general form is:

15. Variance Dispersion Graphs
Prediction variances can be examined with variance dispersion graphs (VDGs), first introduced by Giovannitti-Jensen, A. and Myers, R. H. (1989)
They plot contains max. variance, average variance, and min. variance versus the distance from the center of the design space
Piepel, G., Anderson, C. M, and Redgate, P. E (1993) extended the use of VDGs to irregular regions
VDGs allow us to compare designs to see which have the best overall variance properties over the entire design space

16. Shrinkage

17. VDG - Example

18. Three-Dimensional VDGs
Plot the prediction variance by the distances from the centers of the mixture and process spaces
Distances of 0 signify the center while distances of 1 represent the edges of the spaces
These graphs allow comparisons among designs and evaluation of the relative increases in prediction as the experimenter moves along both spaces, mixture and process
Can be used to assess the optimum placement for additional runs
For the robust design setting, the plots can be done for varying k2a and k2b levels

19. Fish Patty Example
Consider the fish patty example with 3 fish types, deep fry time, baking time, and baking temperature
Initially, consider all of the process variables as controllable, giving the following 28-term model:

20. Fish Patty Designs Six designs of sizes 28 to 40 runs are considered
Two designs are from Cornell and Gorman and were constructed to minimize the size of the confidence intervals of the model coefficients
The other 4 designs are from Design-Expert and were constructed to be D-efficient
The DX6 designs with an “A” had the extra degrees of freedom split between lack-of-fit and replication
The other DX6 designs had all of the degrees of freedom allocated to lack-of-fit

21. Fish Patty 3D VDGs - Surface

22. Fish Patty 3D VDGs - Contours

23. Fish Patty 3D VDGs - Surface

24. 3D VDG - Example
Consider a setting with three mixture components and two controllable process variables (Kowalski, Cornell, and Vining (2000)) with the following 15-term model:
We consider six competing designs – two with 17 runs, three with 23 runs, and one with 25 runs
Extra runs beyond those for model fit were allocated differently for each design

25. Surface Plot VDGs

26. Contour Plot VDGs

27. Example with Noise Variables - Fish Patties
Now consider the fish patty example where baking time and baking temperature are treated as noise variables
We will fit the following 36-term model

28. Fish Patty Designs Three designs are being considered
Design A - The 56-run design from Cornell (2002) which is a 7-run simplex-centroid in the mixture components crossed with an 8-run full factorial in the process and noise variables
Design B - A 56-run design generated with the D-optimal design generator in Design-Expert 6.0 with 10 lack-of-fit points and 10 pure replicate points
Design C - A 36-run design generated with the D-optimal design generator in Design-Expert 6.0 with no lack-of-fit points or pure replicate points
The 3D VDGs show that Design C is the best from a scaled prediction variance standpoint

29. 3D VDGs – Mean Model

30. 3D VDGs – Mean Model

31. 3D VDGs – Mean Model

32. Conclusions
3D VDG’s allow an experimenter to look at the prediction variance profiles of designs in both the mixture and process spaces simultaneously
The plots can be used to compare designs and to determine the placement of additional runs
Noise variables can be handled by looking at a grid of plots

33. References
Borror, C. M., Montgomery, D. C., and Myers R. H. (2002) “Evaluation of Statistical Designs for Experiments Involving Noise Variables”. Journal of Quality Technology 34, pp
Cornell, J. A. (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, Third Edition. John Wiley & Sons, New York, NY.
Giovannitti-Jensen, A. and Myers, R. H. (1989). “Graphical Assessment of the Prediction Capability of Response Surface Designs”. Technometrics 31, pp
Goldfarb, H. B., Borror, C. M., and Montgomery, D. C., (2003). “Mixture-Process Variable Experiments with Noise Variables”. To Appear in the Journal of Quality Technology.
Kowalski, S. M., Cornell, J. A., and Vining, G. G. (2000). “A New Model and Class of Designs for Mixture Experiments with Process Variables”. Communications in Statistics - Theory and Methods 29, pp
Piepel, G., Anderson, C. M, and Redgate, P. E. (1993). “Response Surface Designs for Irregularly-Shaped Regions” (Parts 1, 2, and 3). Proceedings of the Section on Physical and Engineering Sciences, American Statistical Association, Alexandria, Virginia, pp
Steiner, S. H. and Hamada, M. (1997). “Making Mixtures Robust to Noise and Mixing Measurement Errors”. Journal of Quality Technology 29, pp