Three-Dimensional Variance Dispersion Graphs for Mixture-Process Experiments with Control and Noise Variables

1. 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

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

3. 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

4. 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

5. 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

6. 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?

7. Mixture Components xi i = 1, 2, …, q Controllable Process Variables wpp = 1, 2, …, c Noise Variables ztt = 1, 2, …, n Mixture-Process-Noise Model where V is a cn x n block diagonal matrix with w’s on the diagonals and 0’s elsewhere

8. Expected Value and Variance Using the Delta Method:

9. 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:

10. 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

11. 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

12. C-Matrix

13. 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.

14. 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

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

16. 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

17. Shrinkage

18. VDG - Example

19. 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

20. 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:

21. 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

22. Fish Patty 3D VDGs - Surface

23. Fish Patty 3D VDGs - Contours

24. Fish Patty 3D VDGs - Surface

25. 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

26. Surface Plot VDGs

27. Contour Plot VDGs

28. 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

29. 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

30. 3D VDGs – Mean Model

31. 3D VDGs – Mean Model

32. 3D VDGs – Mean Model

33. 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

34. 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. 54-70. • 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. 159-171. • 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. 2255-2280. • 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. 205-227. • Steiner, S. H. and Hamada, M. (1997). “Making Mixtures Robust to Noise and Mixing Measurement Errors”. Journal of Quality Technology 29, pp. 441-450.