FEADO: Program for Constructing D-Optimal Fractional Factorial and Response Surface Designs

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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 window will pop up. Enter 3 at the 2-level factors field and 1 as the 3-level factors field, the FEADO window will become:

and click Start, the following window will pop up:

There are three models: linear, interaction and quadratic. Choose Quadratic as the model and click OK, the following window will pop up:

Choose 12 as the number of runs and click OK, the FEADO window will become:

and the OUTPUT window (not shown) displaying the best constructed design will pop up.

Note:

1. The three model 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 main-effect terms, 2fi terms and squared terms (for 3-level factors).

2. The default random seed is the one obtained from the system clock and the default number of tries is 100. You can change these default values if you wish to.

The following example shows how to construct 5-component mixture design in 25 runs from a candidate set of 269 runs (Example 13). Assuming that you have a file plastic.txt in the working directory which contains candidate runs (Example 14). To make use of this input file, enter plastic.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 quadratic as the model and 15 as the number of runs, you will see the following results in the MIGA output screen:

Note that since the candidate set in the file plastic.txt consists of runs such that ∑x_i=1 where x_i'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 will appear in the OUPUT 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 random seed used;
3. The number of iterations;
4. det=|X'X| where X is the expanded design matrix;
5. det*=|X'X|^1/p/n where n is the number of runs and p is the number of parameters in the model.
6. det^-1;
7. Trace(V) where V=(X'X)^-1;
8. v_max, the maximum prediction variance over N candidate points. The prediction variance v_i at point i (i=1,...N) is calculated as x_i'Vx_i where x_i' is the candidate row i. If the |X'X| values of two competing designs match, the one with a smaller value of v_max will be selected;
9. v_ave, the average prediction variance over N candidate points;
10. G-efficiency defined as 100 p/(n v_max);
11. Design points and variance of the fitted response;
12. X'X;
13. V;
14. 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/s1.html).
2. A 2-level FFD for 10 factors in 11 runs (http://designcomputing.net/gendex/feado/s2.html).
3. A saturated 2-level FFD for 15 factors in 19 runs (http://designcomputing.net/gendex/feado/s3.html).
4. A saturated 2-level FFD of resolution V for 5 factors in 16 runs (http://designcomputing.net/gendex/feado/s4.html).
5. A saturated 2-level FFD of resolution V for 6 factors in 22 runs (http://designcomputing.net/gendex/feado/s5.html).
6. A saturated RSD for 4 factors in 15 runs (http://designcomputing.net/gendex/feado/r1.html).
7. A saturated RSD for 5 factors in 21 runs (http://designcomputing.net/gendex/feado/r2.html).
8. A saturated RSD for 6 factors in 28 runs (http://designcomputing.net/gendex/feado/r3.html).
9. A saturated RSD for 7 factors in 36 runs (http://designcomputing.net/gendex/feado/r4.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:

• Example 3: This design (|X'X|=1.5759E20) improves the design of the same size in Hardin & Sloane (1993), p. 363 (|X'X| = 1.5447E20).
• Example 4: See the corresponding design in Box et al. (1978) Section 12.2.
• Example 5: This design improves the optimal balanced FFD of resolution V of the same size of Srivastava & Chopra (1971) in the sense that trace V of the former is 1.1517 and of the later is 1.6249. See Nguyen & Miller (1992) for detailed comparison of the two types of designs.
• Example 10: The constraints for the design problem are -1≤x_i≤1 and -1.5≤x₁+x₂≤1 (Cf. Mongomery (2001), Section 11-4.4). FEADO uses the design points in Table 11-12 of this reference as candidate points (adhesive.txt) and decreases the |X'X|^-1 of this design (using the quadratic model) from 2.153E-4 to 1.8156E-4.
• Example 11: The constraints used to construct the candidate set are Σx_i=1, 0.05≤x₁≤0.25, 0.25≤x₂≤0.40 and 0.50≤x₃≤0.70. (Cf. Mongomery (2001), Example 11-4). FEADO uses the design points in Table 11-14 of this reference as candidate points (paint.txt) and decreases the |X'X|^-1 of this design from 3.2422E13 to 1.8335E13.
• Example 12: The constraints used to construct the candidate set are Σx_i=1, 0.00≤x₁≤0.10, 0.00≤x₂≤0.10, 0.05≤x₃≤0.15, 0.20≤x₄≤0.40 and 0.40≤x₅≤0.60 (Cf. Snee & Marquardt (1974)). FEADO selects 16 points from a candidate set consisting 28 vertices (gasoline.txt) using linear model. The resulting design has |X'X|^-1= 13807 and the corresponding design constructed by Heredia-Langner, et.al. (2003) using the genetic algorithm with |X'X|^-1=13827.
• Example 13: The constraints used to construct the candidate set are Σx_i=1, 0.50≤x₁≤0.70, 0.05≤x₂≤0.15, 0.05≤x₃≤0.15, 0.10≤x₄≤0.25. and 0.00≤x₅≤0.15. The additional constraints are 0.18 ≤x₄+x₅≤0.26 and x₃+x₄+x₅≤0.35 (Cf. Snee (1985), p. 233). FEADO selects 25 points from a candidate set consisting 269 vertices, edge centers, check runs and overall centroid (plastic.txt) using the quadratic model. This candidate set was constructed by Design Expert Version 6 from Stat Ease (http://www.statease.com). The resulting design has |X'X|^-1=1.2162E48.
• Example 14: The constraints are the same as the one used for the previous example. The candidate set was however, a grid of step size 0.01 in all variables which consists of 10,468 design points (plastic2.txt). The resulting 25-point design obtained from this candidate set has |X'X|^-1=1.187E48 and the corresponding design constructed by Heredia-Langner, et.al. (2003) has |X'X|^-1=1.217E48.

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, (5^th 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). 2^m 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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