# CUT: Program for Multi-Dimensional Blocking Fractional Factorial and Response Surface Designs

## Introduction

CUT is a Gendex module for multi-dimensional blocking fractional factorial designs (FFDs) and response surface designs (RSDs). The CUT approach to blocking a design is to find a suitable unblocked design and allocate the n runs of this design to blocks, or rows and columns, etc. such that the objective function f is minimized. f is defined such that when the design is orthogonally blocked, f becomes 0. The algorithm which implements the CUT approach is the extension of the one appeared in Nguyen (2001). The 2015 version of CUP is modified to block definitive screening designs (DSDs) with augmented 2-level factors. In this note Wu & Hamada (2000) is abbreviated as WH and Box-Behnken designs of Box & Behken (1960) are abbreviated as BBDs.

## Using CUT

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to divide a DSD with four 3-level factors and four 2-level factors (Jones & Nachtsheim, 2013) into three rows and three columns. This unblocked design is in the file DSD4.txt in the working directory:

At the working directory, type the following command at the command prompt (case is important):

`java -cp c:\gendex CUT`

The CUT GUI will pop up. Enter DSD4.txt at the File text field and 2 as the number of blocking factors at 3-level. The CUT GUI will become: 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 screen will pop up: There are four model options: (i) Linear; (ii) Interaction; and (iii) Quadratic and (iv) Pure-quadratic. Choose Pure-quadratic, CUT will start running and stop after 1,000 tries. The plan for the constructed blocked design for the best try will then pops up in the CUT output window: The START button has been changed to the RESET one. If you click this RESET button, the output will disappear and you can use CUT for a new design problem.

We now modify the definition of clear and strongly clear of an effect in WH p. 181 to suite our explanation of model options. A main effect or 2-factor interaction (2fi) is clear if it is orthogonal to other main effects, 2fi's and block effects. A main effect is strongly clear if it is clear and any 2fi's involving it is clear. An 11-factor design with the first three strongly clear factors, for example should have the following clear 2fi's: 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 23, 24,25, 26, 27, 28, 29, 2a, 2b, 34, 35, 36, 37, 38, 39, 3a, 3b (Example 13). Here, we use the hexadecimal system to represent the factor, i.e. 10 is represented by 'a' and 11 by 'b', etc.

The four model options are:

1. Linear: includes only the main-effect terms.
2. Interaction: includes all main-effect terms and all 2-factor interaction (2fi) terms;
3. Quadratic: includes all main-effect terms, all 2fi terms and squared terms (for 3-level factors).
4. Pure-quadratic: includes all main-effect terms and squared terms (for 3-level factors).

When option Interaction is chosen, you have to set a value for m', a preset number of strongly clear main effects. Set m' to 0 if you want all main-effects clear and to a value from 1 to m-1 if you want m main-effects clear and the first m' main-effects strongly clear. CUT has been used to construct the designs in Table 1. This table shows the block factor (BF) values of designs for m factors and n runs in b blocks. The values of b's are in brackets. BF is a measure of orthogonality of a blocked design (Nguyen, 2001). BF equals 1 means the design is orthogonally blocked.

Table 1. BF values of constructed designs
m n=8 n=16 n=32 n=64
3 1.0000 (2) - - -
4 - 1.0000 (2)
0.882 (4)
- -
5 - - 1.0000 (2)
1.0000(4)
0.841 (8)
-
6 - - 1.0000 (2)
0.939 (4)
0.828 (8)
1.0000 (2)
1.0000 (4)
1.0000 (8)
7 - - - 1.0000 (2)
1.0000 (4)
1.0000 (8)
8 - - - 1.0000 (2)
1.0000 (4)
0.880 (8)

Note that when the unblocked design consists of runs from a mixture experiment, you will only have two options for model: (i) linear and (ii) quadratic.

## Output

The result of the best try is displayed in the CUT output window and is also saved in the file CUT.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. The objective function f . The session automatically stops if f becomes 0, i.e. when the design becomes orthogonally blocked. If option Interaction is chosen and m', CUT will use two objective functions f and g. g=0 indicates that a design with m clear main effects (or with m clear main effects and the first m' strongly clear main effects) is obtained.
4. trace of (X'X)-1;
5. The standardized determinant |X'X|1/p/n ;
6. BF value;
7. Plan of the blocked design and the associated random seed;
8. The X matrix;
9. The X'X matrix;
10. (X'X)-1;
11. The time in seconds CUT used to construct the above design;

## Examples

1. Divide a 23 factorial into two blocks (http://designcomputing.net/gendex/cut/f1.html).
2. Divide a 23 factorial into four blocks (http://designcomputing.net/gendex/cut/f2.html).
3. Divide a 24 factorial into four blocks (http://designcomputing.net/gendex/cut/f3.html).
4. Divide a 25 factorial into eight blocks (http://designcomputing.net/gendex/cut/f4.html).
5. Divide a 26-1 fractional factorial into four blocks (http://designcomputing.net/gendex/cut/f5.html).
6. Divide a 26-1 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/f6.html).
7. Divide a 28-2 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/c7.html).
8. Divide 18 runs (which includes a 24 factorial) into three blocks (http://designcomputing.net/gendex/cut/f8.html).
9. Divide a 33 factorial into three rows and three columns (http://designcomputing.net/gendex/cut/f9.html).
10. Divide the 3-factors BBD into three blocks (http://designcomputing.net/gendex/cut/b3.html).
11. Divide the 4-factors BBD into three blocks (http://designcomputing.net/gendex/cut/b4.html).
12. Divide the 4-factors BBD into two rows and two columns (http://designcomputing.net/gendex/cut/b4bis.html).
13. Divide the 5-factors small BBD of Pham & Nguyen (2014) into two rows and two columns (http://designcomputing.net/gendex/cut/b5.html).
14. Divide the design 6-factors small BBD of Pham & Nguyen (2014) into two rows and two columns (http://designcomputing.net/gendex/cut/b6.html).
15. Divide the design D736 of Nguyen & Borkowski (2008) design into two rows and two columns (http://designcomputing.net/gendex/cut/b7.html).
16. Divide a 3-factor central-composite design of Box & Draper (1987, p. 360) design into four blocks (http://designcomputing.net/gendex/cut/BH.html).
17. Divide 6 distinct binary blends into two blocks (http://designcomputing.net/gendex/cut/m1.html).
18. Divide 24 distinct 4-component blends into two two rows and two columns (http://designcomputing.net/gendex/cut/m2.html).
19. Divide 16 distinct binary blends into two two rows and two columns (http://designcomputing.net/gendex/cut/m3.html).
20. Divide 24 distinct 3-component blends into two two rows and two columns (http://designcomputing.net/gendex/cut/m4.html).
21. Divide a DSD with five 3-level factors into three blocks (http://designcomputing.net/gendex/cut/d1.html).
22. Divide an augmented-DSD with four 3-level factors and seven 2-level factors into three blocks (http://designcomputing.net/gendex/cut/d2.html).
23. Divide a augmented-DSD with four 3-level factors and three 2-level factors into three rows and three columns (http://designcomputing.net/gendex/cut/d3.html).
24. Divide a augmented-DSD with four 3-level factors and four 2-level factors into three rows and three columns (http://designcomputing.net/gendex/cut/d4.html).

Notes:

• Example 2: See the corresponding design in WH Table 3.A. Each block of the blocked design forms a fold-over pair.
• Examples 3-7: The main-effects of the designs in these examples are clear. Some designs also have the first few main-effects strongly clear. Unlike the corresponding designs in WH Table 3.A and 4B, these designs do not completely confound any 2fi with blocks. As such, they are more useful in general settings where the experimenters are not prepared to sacrifice any 2fi.
• Example 8: See Nguyen (2001) Example 2. The unblocked design for this example consists of the 24 factorial and a fold-over pair (1) and abcd . All main effects are orthogonal to block effects. See an alternative solution in Cook & Nachtsheim (1989) Section 3.2.
• Examples 9-16: Examples of orthogonally blocked row-column response surface designs.
• Example 17: See Cornell (1990), p. 438. It is conventional to add a centroid blend to each block. In this example the centroid blend is (0.3333, 0.3333, 0.3333). To analyse this design and the constructed design in the next four examples, reduce the number of block terms by one (See Cornell, 1990 or Draper et al ., 1993).
• Example 18: See Cornell (1990), p. 439.
• Example 19: See Table 2 of Draper et al ., 1993.
• Example 20: See Table 2 of Nguyen (2001). The unblocked design consists of 24 distinct 3-component blends (0.00, 0.05, 0.25, 0.70), etc.
• Example 21-24: The unblocked DSDs with augmented 2-levels factors in these examples are either from Jones & Nachtsheim (2013) or Nguyen & Pham (2016).

## References

Box, G.E.P. & Behnken, D.W. (1960) Some new three-level designs for the study of qualitative variables. Technometrics 2, 455-475.
Box, G.E.P. & Draper, N.R. (1987) Empirical model building and Response Surfaces (New York, Wiley).
Cornell, J.A. (1990) Experiments with mixture designs, models and the analysis of mixture data. 2nd ed. New York: John Wiley & Sons,Inc.
Cook, R.D. & Nachtsheim, C.J. (1989) Computer-aided blocking of factorial and response surface designs. Technometrics 31, 339-346.
Draper, N.R., Prescott, P., Lewis, S.M., Dean, A.M., John, P.W.M & Tuck, M.G. (1993) Mixture designs for four components in orthogonal blocks. Technometrics 35, 268-276.
Pham D-T & Nguyen, N-K (2014) Small Box-Behnken Designs With Orthogonal Blocks,. Statistics & Probability Letters 85, 84-90.
Jones, B., & Nachtsheim, C. J. (2013). Definitive Screening Designs with Added Two-Level Categorical Factors. Journal of Quality Technology, 45, 121-129.
Nguyen, N-K (2001) Cutting experimental designs into blocks. Austral. & New Zealand J. of Statistics 43, 367-374.
Nguyen, N-K & J.J. Borkowski (2008) New 3-level response surface designs constructed from incomplete block designs. J. of Statistical Planning & Inference 138, 294-305.
Nguyen, N-K & Pham T-D (2016) Small Mixed-Level Screening Designs with Orthogonal Quadratic Effects, Journal of Quality Technology, 48, 405-414.
Wu, C.F.J & M. Hamada (2000) Experiments: Planning, Analysis and Parameter Design Optimization. New York: John Wiley & Sons, Inc.

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