FEADO: Program for Constructing D-Optimal Designs by the Fedorov Algorithm

  1. Introduction
  2. Using FEADO
  3. Output
  4. Examples
  5. References

Introduction

FEADO (Fedorov exchange algorithm for D-optimal experimental designs) constructs D- and G-optimal (and near-optimal) 2-level and mixed-level fractional factorial designs (FFDs) and response surface designs (RSDs). FEADO can also construct designs for constrained regions including mixture designs by choosing a subset of runs from a given set of candidate runs (Examples 13-15). There is no limit on the number of runs in the candidate set.

FEADO uses a fast Fedorov's exchange algorithm described in Nguyen & Miller (1992), Miller & Nguyen (1994) and Nguyen & Piepel (2004).

Using FEADO

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct a saturated design for three factor each at 2-level and one factor at 3-level. At the working directory, type the following command at the Command Prompt (case is important):

java -cp c:\gendex FEADO

The FEADO GUI will pop up. Enter 3 in the At 2-level (factor) field and 1 in the At 3-level (factor) field, the FEADO GUI will become:

Now click Start, the following window will pop up:

There are four models: Linear, Interaction, Quadratic and Pure-quadratic . Click Quadratic, the following window will pop up:

Choose 12 as the number of runs and click OK, FEADO will start running and in less than a second the plan for the design constructed in try 2 pops up. This is the best design out of 1000 constructed designs (which correspond to 1000 tries):


Note:

1. The four four options are: (i) Linear: only includes the main-effect terms; (ii) Interaction: includes the main-effect terms and 2-factor interaction (2fi) terms; and (iii) Quadratic: includes the squared terms (for 3-level factors), the main-effect terms and 2fi terms; and (iv) Pure-quadractic: includes the squared terms (for 3-level factors), the main-effect terms.

2. The default random seed is the one obtained from the system clock and the default number of tries is 1000. You can change these default values if you wish to.

The following example shows how to construct 5-component mixture design in 16 runs from a candidate set of 28 runs (Example 12). Assuming that you have a file gasoline.txt in the working directory which contains candidate runs. To make use of this input file, enter gasoline.txt in the File text field. The number of factors and runs will appear in the Factor and Runs field. At the same time the Number of 2-level factors and Number of 3-level factors fields will become uneditable. Now click Start and choose Linear as the model and 16 as the number of runs, you will see the following results in the FEADO output screen:

Note that since the candidate set in the file plastic.txt consists of runs such that ∑xi=1 where xi's are the components of the mixture experiment, you can only choose one of the following model options: (i) Linear and (ii) Quadratic.

Output

The result of the best try is displayed in the FEADO output window and is also saved in the file FEADO.htm in the working directory. This file can be read by a browser such as IE or Firefox. Information for this try includes:

  1. Try number;
  2. The number of iterations;
  3. det=|X'X| where X is the expanded design matrix;
  4. det-1;
  5. trace of (X'X)-1;
  6. vmax, the maximum prediction variance over N candidate points. The prediction variance vi at point i (i=1,...N) is calculated as xi'Vxi where xi' is the candidate row i. If the |X'X| values of two competing designs match, the one with a smaller value of vmax will be selected;
  7. vave, the average prediction variance over N candidate points;
  8. G-efficiency defined as 100 p/(n vmax);
  9. ln(|X'X|);
  10. Design points and variance of the fitted response and the associated random seed;
  11. X'X and (X'X)-1;
  12. The time in seconds FEADO used to construct the above design.

Examples

  1. A saturated 2-level FFD for 11 factors in 12 runs (http://designcomputing.net/gendex/feado/l1.html).
  2. A 2-level FFD for 10 factors in 11 runs (http://designcomputing.net/gendex/feado/l2.html).
  3. A saturated 2-level FFD for 15 factors in 19 runs (http://designcomputing.net/gendex/feado/l3.html).
  4. A saturated 2-level FFD of resolution V for 5 factors in 16 runs (http://designcomputing.net/gendex/feado/l4.html).
  5. A saturated 2-level FFD of resolution V for 6 factors in 22 runs (http://designcomputing.net/gendex/feado/l5.html).
  6. A saturated RSD for 4 factors in 15 runs (http://designcomputing.net/gendex/feado/q4.html).
  7. A saturated RSD for 5 factors in 21 runs (http://designcomputing.net/gendex/feado/q5.html).
  8. A saturated RSD for 6 factors in 28 runs (http://designcomputing.net/gendex/feado/q6.html).
  9. A saturated RSD for 7 factors in 36 runs (http://designcomputing.net/gendex/feado/q7.html).
  10. A 2-factor RSD for the constrained region in 12 runs (http://designcomputing.net/gendex/feado/adhesive.html).
  11. A 3-component mixture design in 14 runs (http://designcomputing.net/gendex/feado/paint.html).
  12. A 5-component mixture design in 16 runs (http://designcomputing.net/gendex/feado/gasoline.html).
  13. A 5-component mixture design in 25 runs (http://designcomputing.net/gendex/feado/plastic.html).
  14. A 5-component mixture design in 25 runs (http://designcomputing.net/gendex/feado/plastic2.html).

Notes:

References

Box, G. E. P, Hunter, W. G. & Hunter, J. S. (1978). Statistics for experimenters. New York: John Wiley.
Hardin, R. H. & Sloane, N. J. A. (1993) A new approach to the construction of optimal designs. Statistical Planning & Inference 37, 339-369.
Heredia-Langner, A., Carlyle, W. M., Mongomery, D. C., Borror & C. M., Runger, G.C. (2003) Genetic algorithm for the construction of D-optimal designs. Journal of Quality Technology 35, 28-46.
Miller A. J. & Nguyen, N-K. (1994). A Fedorov exchange algorithm for D-optimal designs. Applied Statistics 43, 669-678.
Mitchell, T. J. (1974). An algorithm for the construction of D-optimal designs. Technometrics 16, 203-210.
Mongomery, D. C (2001). Design and analysis of experiments, (5th edition). New York: John Wiley.
Mongomery, D. C., Loredo, E. N., Jearkpaporn D., & Testik, M. C. (2002) Experimental designs for constrained regions. Quality Engineering 14, 587-601.
Nguyen, N-K. & Miller A. J. (1992). A review of some exchange algorithms for constructing discrete D-optimal designs. Computational Statist. & Data Analysis 14, 489-498.
Nguyen, N-K. & Miller A. J. (1996). 2m fractional factorial designs of resolution V with high A-efficiency, 7≤m≤10. J. Statistical Planning & Inference 59, 379-384.
Nguyen, N-K & Piepel G. F. (2005) Computer-generated experimental designs for irregular-shaped regions. Quality Technology & Quantitative Management 2, 147-160.
Snee, R. D. & Marquardt, D.W. (1974). Extreme vertice designs for linear mixture models.. Technometrics 16, 399-408.
Snee, R. D. (1985). Computer-aided design of experiments--Some practical experiences. Journal of Quality Technology 17, 222-236.
Srivastava, J.N. & Chopra, D.V. (1971). Balanced optimal 2m fractional factorial design of resolution V, m≤6. Technometric 13, 257-269.

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