# MIGA: Program for Constructing Minimum G-Aberration Designs

## Introduction

MIGA is a program for constructing minimum G-aberration designs. The minimum G-aberration criterion proposed by Tang & Deng (1999) is a generalized version of the popular minimum aberration criterion of Fries & Hunter (1980). Designs constructed by MIGA include both regular and non-regular 2-level fractional factorial designs.

Let Dmxn be a design with m columns (factors) c1,c2,...,cm each of size n with n/2 +1's and n/2 -1's. We now use each row i of D to construct a vector Ji of length mC2+mC3+mC4. Define the first mC2 element as ci1ci2, ci1ci3,... the next mC3 as ci1ci2ci3, ci1ci2ci3, ... and the last mC4 as ci1ci2ci3ci4, ci1ci2ci3ci5 ...where cij is the ith entry of column cj. Let J=∑ Ji.

Now, define B2(D), B3(D) and B4(D) as 1/n2 x the sum of squares of of the first mC2, the next mC3 and the last mC4 elements of J respectively (see Eq. 2 of Ingram & Tang, 2005). Therefore, B2(D) provides the extent of aliasing between main effects in D, B3(D) provides the extent of aliasing between main effects and 2-factor interactions in D, and B4(D) provides the extent of aliasing between pairs of 2-factor interactions in D (see Ingram & Tang, 2005)

A design D is said to have less G2 aberration than D' if

1. B2(D)<B2(D') or
2. B2(D)=B2(D') and B3(D)<B3(D') or
3. B2(D)=B2(D') and B3(D)=B3(D') and B4(D)<B4(D').

MIGA starts with a random design D with m columns, each having n/2 +1's and n/2 -1's. It then continues improving D with repect to the minimum G2 aberration criterion by exchanging the positions of +1 and -1 in each column of D. Details of the MIGA algorithm will be reported elsewhere.

In this note Wu & Hamada (2000) is abbreviated as WH.

## Using MIGA

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct a 27-2 resolution VI design. At the working directory, type the following command at the command prompt (case is important):
`java -cp c:\gendex MIGA`

The MIGA window will pop up. Enter 7 as the the number of 2-level factors and 32 as the number of runs:

Note that 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. Now, click START, the following window will pop up:

Choose 3 (see Table 1 below) and click OK, MIGA will start running and in less than a second, the 27-2 resolution IV design will pop up in the MIGA output window:

Note that the START button has been changed to the RESET one. If you click this RESET button, the output will disappear and you can use MIGA for a new design problem.

We now modify the definition of clear and strongly clear of an effect in WH p. 156 to suite our explanation of the resolution options. A main effect or 2-factor interaction (2fi) is clear if it is orthogonal to other main effects and 2fi's. A main effect is strongly clear if it is clear and any 2fi's involving it is clear. A 7-factor design with the first three strongly clear factors 1-3, for example, should have the following 15 clear 2fi's: 12, 13, 14, 15, 16, 17, 23, 24, 25, 26, 27, 34, 35, 36 and 37 (Example 3).

A constructed design will be classified into the following resolution type:

1. III: In a resolution III design, main effects are pair-wise orthogonal. Resolution III design can be constructed more efficiently with the NOA module of the Gendex DOE toolkit.
2. V: In a resolution V design, all main effects are strongly clear.
3. IV (with the first m' main effects strongly clear): In a resolution IV design, all main effects are clear. However, the numbers of m' (<m), the first strongly clear main effects of two resolution IV designs of the same size might differ. To construct the resolution IV designs for 6≤m≤11, and n=16, 32 and 64 (Examples 3-9), you have to use Table 1. In this table, the preset values of m' are given for each value of m and n. The numbers in the brackets in this table refer to the numbers of clear 2fi's associated with each value of m'. Note that for m=9 and n=32, you have two choices of m' (Examples 5-6).

Table 1. Preset values of m'
m n=16 n=32 n=64
6 0 (0) - -
7 0 (0) 3 (15) -
8 0 (0) 2 (13) -
9 - 1 (8)
2 (15)
5 (30)
10 - 0 (0) 4 (30)
11 - 0 (0) 3 (27)

MIGA can also augment additional columns to existing design. To construct a 28-2 resolution VI design, you can add an additional column to the 27-2 design constructed previously which is now used as a base design:

## Output

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

1. Try number;
2. The number of iterations;
3. B2(D), B3(D) B4(D) values. MIGA automatically stops when all three values become 0.
4. The design plan and the associated random seed.
5. The non-zero elements in J if any.
6. Resolution of the design.
7. A list of clear 2fi's if the design is of resolution IV.
8. The time in seconds MIGA used to construct the above design.

## Examples

1. A 25-1 resolution V design (http://designcomputing.net/gendex/miga/f16x5.html).
2. A 26-2 resolution IV design (http://designcomputing.net/gendex/miga/f16x6.html).
3. A 27-2 resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x7.html).
4. A 28-3 resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x8.html).
5. A 29-4 resolution IV design with 1 strongly clear main effect (http://designcomputing.net/gendex/miga/f32x9.html).
6. A 29-4 resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x9bis.html).
7. A 29-3 resolution IV design with 5 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x9.html).
8. A 210-4 resolution IV design with 4 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x10.html).
9. A 211-3 resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x11.html).
10. A 2-level design for 4 factors in 12 runs (http://designcomputing.net/gendex/miga/h12x4.html).
11. A 2-level design for 6 factors in 24 runs (http://designcomputing.net/gendex/miga/h24x6.html).
12. A 2-level for 15 factors in 16 runs (http://designcomputing.net/gendex/miga/h16x15.html).
13. A 2-level design for 23 factors in 24 runs (http://designcomputing.net/gendex/miga/h24x23.html).
14. A 2-level design with minimal aliasing for 6 factors in 12 runs (http://designcomputing.net/gendex/miga/h12x6bis.html).

Notes:

• Example 1-2: The first 4 factors of these designs (generated by MIGA) make up a 24 factorial.
• Example 3: The first 5 factors of these designs (generated by MIGA) make up a 25 factorial. See the corresponding design in WH Table 4.A3. Both designs have 3 strongly clear main effects and 15 clear 2fi's.
• Example 4: Obtained by adding a 2-level factor to the design in Example 3. See the corresponding design in WH Table 4.A3. Both designs have 2 strongly clear main effects and 13 clear 2fi's.
• Example 5: Obtained by adding a 2-level factor to the design in Example 4. See the corresponding design in WH Table 4.A3. Both designs have 1 strongly clear main effects and 8 clear 2fi's.
• Example 6: Obtained by adding a 2-level factor to the design in Example 4. See the corresponding design in WH Table 4.A3. Both designs have 2 strongly clear main effects and 15 clear 2fi's. There is a small price to pay for maximizing the number of strongly clear main effects. While the J vector of this design has 7 non-zero elements or 21 pairs of non-orthogonal 2fi's, the one of the design in Example 5 (also called minimum aberration design) has only 6 non-zero elements or 18 pairs of non-orthogonal 2fi's. See WH Section 4.5 for the minimum aberration and related criteria in design selection.
• Example 7: Obtained by adding a 2-level factor to a 28-2 resolution V design. See the corresponding design in WH Table 4.A5. Both designs have 5 strongly clear main effects and 30 clear 2fi's.
• Example 8: Obtained by adding a 2-level factor to the design in Example 7. See the corresponding design in WH Table 4.A5. The MIGA design has 4 strongly clear main effects and 30 clear 2fi's. The WH design has 2 strongly clear main effects and 33 clear 2fi's.
• Example 9: Obtained by adding a 2-level factor to the design in Example 7. See the corresponding design in WH Table 4.A5. The MIGA design has 3 strongly clear main effects and 27 clear 2fi's. The WH design has 1 strongly clear main effects and 34 clear 2fi's.
• Examples 10 and 11: These designs corresponds to designs IF0412 and IF0624 in Haaland (1989).
• Example 12: This design is constructed sequentially. The first 4, 5,...14 columns of this design form designs for m=4, 5,...15 and correspond to those in Table 2 of Tang & Deng (1999).
• Example 13: This design is constructed sequentially. The first 4, 5,...22 columns of this design form designs for m=4, 5,...22 and correspond to those in Tables 1 and 2 of Ingram & Tang (2005). Note that for m≤12, both B2(D) and B3(D) values of MIGA designs are 0. For m>12, although only B2(D) values of the MIGA designs are 0, none of them has the 2fi's totally confounded.
• Example 15: This is an an example of a design with minimal aliasing constructed by using option Minimal aliasing (Cf. the design in the file h12x6.html and the one in Table 2 of Jones & Nachtsheim, 2011)

## References

Fries, A. & Hunter, W. G. (1980) Minimum Aberration 2k-p designs. Technometrics 22, 601-608.
Ingram, D. & Tang, B. (2005) Minimum G Aberration design construction and design tables for 24 runs. J. of Quality Technology 37, 101-114.