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

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).

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).

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≤xi≤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 ∑xi=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 ∑xi=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 ∑xi=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

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