## 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 *D _{mxn}* be a design with

*m*columns (factors)

*c*each of size

_{1},c_{2},...,c_{m}*n*with

*n*/2 +1's and

*n*/2 -1's. We now use each row

*i*of

*D*to construct a vector

*J*of length

_{i}

_{m}C_{2}+

_{m}C_{3}+

_{m}C_{4}. Define the first

_{m}C_{2}element as

*c*

_{i1}c

_{i2},

*c*

_{i1}c

_{i3},... the next

_{m}C_{3}as

*c*

_{i1}c

_{i2}c

_{i3},

*c*

_{i1}c

_{i2}c

_{i3}, ... and the last

_{m}C_{4}as

*c*

_{i1}c

_{i2}c

_{i3}c

_{i4},

*c*

_{i1}c

_{i2}c

_{i3}c

_{i5 }...where

*c*

_{ij}is the

*i*th entry of column

*c*. Let

_{j}*J=∑ J*.

_{i}Now, define *B*_{2}(*D*),
*B*_{3}(*D*) and *B*_{4}(*D*)
as
1/*n*^{2} x the sum of squares of of the first
_{m}C_{2}, the next _{m}C_{3}
and the last _{m}C_{4} elements of *J*
respectively (see Eq. 2 of Ingram & Tang, 2005). Therefore,
*B*_{2}(*D*) provides the extent of aliasing
between main
effects in *D*, *B*_{3}(*D*) provides
the extent
of aliasing between main effects and 2-factor interactions in *D*,
and
*B*_{4}(*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 *G*_{2}
aberration
than *D*' if

*B*_{2}(*D*)<B_{2}(*D'*) or*B*_{2}(*D*)=*B*_{2}(*D'*) and*B*_{3}(*D*)*<B*(_{3}*D'*) or*B*_{2}(*D*)=*B*_{2}(*D*') and*B*_{3}(*D*)*=B*(_{3}*D*') and*B*_{4}(*D*)*<B*(_{4}*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 G_{2} 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 2^{7-2}resolution VI design. At the working directory, issue the following command at the command prompt (case is important and no space is allowed before/after the equal sign):

java -cp c:\gendex miga m=7 ms=3 n=32

**miga** will start running and after try 1, the plan of the constructed design for this try will be displayed at the terminal window and then **miga** stops:

miga 10.0: Program for Constructing Minimmum G-Aberration Designs (c) 2018 Design Computing (http://designcomputing.net/) Note: Best design for 7 2-level factors in 32 runs. try iter B2 B3 B4 1 14 0 0 1 Design for try 1 using seed 1534226728132: (1) (2) (3) (4) (5) (6) (7) 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 Note: Non-zero elements in J: 4567 (32) Note: Resolution IV (all main effects and 15 2fi's 12 13 14 15 16 17 23 24 25 26 27 34 35 36 37 are clear). Note: The first 5 column(s) form the base design. Note: miga used 0.205 seconds. Note: miga.htm has been created.

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 by specifying the seed number and the number of tries, e.g.

java -cp c:\gendex miga m=7 ms=3 n=32 seed=1234 tries=1000

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:

- 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.
- V: In a resolution V design, all main effects are strongly clear.
- IV (with the first
*m*main effects strongly clear): In a resolution IV design, all main effects are clear. However, the numbers of_{s}*m*(<_{s}*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_{s}*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_{s}*m*=9 and*n*=32, you have two choices of*m*(Examples 5-6)._{s}Table 1. Preset values of *m*_{s}*m**n*=16*n*=32*n*=646 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
2^{8-2} resolution VI design, you can add an additional column
to the
2^{7-2} design constructed in the previous example which is now used as a base design with the following command:

java -cp c:\gendex miga in=miga.in m=8

## Output

The result of the best try is displayed at the terminal 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:

- Try number;
- The number of iterations;
*B*_{2}(*D*),*B*_{3}(*D*)*B*_{4}(*D*) values.**miga**automatically stops when all three values become 0.- The design plan and the associated random seed.
- The non-zero elements in
*J*if any. - Resolution of the design.
- A list of clear 2fi's if the design is of resolution IV.
- The time in seconds
**miga**used to construct the above design.

## Examples

- A 2
^{6-2}resolution IV design (http://designcomputing.net/gendex/miga/m1.html). - A 2
^{7-2}resolution IV design (http://designcomputing.net/gendex/miga/m2.html). - A 2
^{8-2}resolution IV design (http://designcomputing.net/gendex/miga/m3.html). - A 2
^{7-2}resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/miga/m4.html). - A 2
^{8-3}resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/miga/m5.html). - A 2
^{9-4}resolution IV design with 1 strongly clear main effect (http://designcomputing.net/gendex/miga/m6.html). - A 2
^{9-4}resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/miga/m7.html). - A 2
^{9-3}resolution IV design with 5 strongly clear main effects (http://designcomputing.net/gendex/miga/m8.html). - A 2
^{10-4}resolution IV design with 4 strongly clear main effects (http://designcomputing.net/gendex/miga/m9.html). - A 2
^{11-3}resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/miga/m10.html).

**Notes**:

- Example 1-3: The first 4 factors of these designs (generated by
**miga**) make up a 2^{4}factorial. - Example 4: This example and examples 5-7 were obtained by adding 2-level
factors to a 2
^{5}factorial. See the corresponding design in WH Table 4.A3. Both designs have 3 strongly clear main effects and 15 clear 2fi's. - Example 5: See the corresponding design in WH Table 4.A3. Both designs have 2 strongly clear main effects and 13 clear 2fi's.
- Example 6: See the corresponding design in WH Table 4.A3. Both designs have 1 strongly clear main effects and 8 clear 2fi's.
- Example 7: 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 7 (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 8: This example and example 9 were
obtained by adding 2-level factors to a 2
^{6}factorial. See the corresponding design in WH Table 4.A5. Both designs have 5 strongly clear main effects and 30 clear 2fi's. - Example 9: 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 10: Obtained by adding a 2-level factor to the design in Example
8. 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.

^{References}

Fries, A. & Hunter, W. G. (1980) Minimum Aberration
2^{k-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.

Tang, B. & Deng, L.Y. (1999) Minimum G_{2}-aberration for
nonregular fractional factorial designs. *The Annals of Statistics*
**27**, 1914-1926.

Wu, C.F.J & M. Hamada (2000) Experiments: Planning, Analysis and Parameter
Design Optimization. New York: John Wiley & Sons, Inc.

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