SOD: Program for Constructing Second-order Designs

Gendex Home Gendex Modules Alpha module IBD module CIBD module RCD module RRCD module FEADO module MIGA module CUT module RAT module NOA module LHD module Sudoku module

1. Introduction
2. Using SOD
3. Output
4. Examples
5. References

Introduction

SOD constructs 3-level second-order response surface designs (RSDs) for m factors in n runs D_mxn where each factor takes only levels -1, 0 and +1 (and for each factor the numbers of +1's and -1's are the same). The constructed designs include Box-Behnken designs (BBDs) of Box & Behnken (1960), Central Composite designs (CCDs) of Box & Wilson (1951), Draper-Lin Small Composite Designs (SCDs) of Draper & Lin (1990), etc. SOD can also augment hard-to-change (HTC) factors with additional easy-to-change (ETC) factors. Split-plot RSDs which satisfy the balanced equivalent estimation property expressed in equation (4) of Parker, et al. (2006) can be constructed by this facility (see Example 14-20). SOD can also augment existing runs with additional runs.

Let D be 3-level RSD for m factors in n runs with the same numbers of +1's and -1's. We have Σx_i=0, Σx³_i=0 and Σ x²_i=Σ x⁴_i =b_i where b_i is the number of ±1 of factor i. Let's impose the following conditions on D:

(i) Σx²_ix_j=0 for i<j<k;
(ii) Σx²_ix_jx_k=0 for i<j<k;
(iii) Σ x_ix_j=0 (and x³_ix_j=0) for i<j;
(iv) Σx_ix_jx_k=0 for i<j<k;
(v) Σx_ix_jx_kx_l=0 for i<j<k<l;
(vi) Σx²_ix²_j-b_ib_j/n=0 for i<j.

The summations are taken over the n design points . If D is also orthogonal, the above conditions are satisfied (see John, 1971 Section 10.2).

SOD construct D by starting with a 3-level design and assign an equal number of ±1's to columns of D_mxn,. Let f be the sum of squares of the sums in (i)-(v) and g be the sum of squares of the sums in (vi). SOD then sequentially minimize the objective functions f and g by switching the positions of -1, 0 and +1 in each columns of D. When both f and g becomes 0 (when both conditions are satisfied), D becomes orthogonal. The detailed account of the SOD algorithm appears in Nguyen & Lin (2010). A similar approach has been used by Nguyen (1996a, 1996b) to construct supersaturated designs and near-orthogonal arrays.

Using SOD

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct an RSD for five factors (three at 3-level and two at 2-level) in 36 runs. At the working directory, type the following command at the command prompt (case is important):

java -cp c:\gendex sod

The SOD window will pop up. Enter 2 in the 2-level factors field, 3 in the 3-level factors field and 36 at the factorial runs field. 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 12 and click OK. In less than a second, the output window showing the constructed design pops up and the SOD window becomes:

Note that the START button has been changed to the STOP one. If you close the pop-up window, the STOP button will become a RESET one. If you click this RESET button, the output in the SOD window will disappear and you can use SOD for a new design problem.

To construct the BBD-type designs for 3-7 factors (Examples 4-8) you have to use the preset values of n_f (factorial runs from a 3^m factorial), n₀ (center runs) and 0's (number of 0-level for each 3-level factor) in Table 1:

Table 1. Preset values of n_f, and n₀ and 0's

m	n	nf	n0	0's
3	15	12	3	5
4	27	24	3	13
5	46	40	6	24
6	54	48	6	24
7	62	56	6	32

To construct the small composite designs of Draper &Lin (1990) in examples 9-13 or the central-composite designs of Box & Wilson (1951) and you have to enter a file which include the n_a axial runs in the File field, click Runs in the Add additional panel, specify appropriate number runs n_f in the < strong">Factorial runs(and Center runs) field using the following preset values of n_a and n_f in Table 2:

Table 2. Preset values of n_f

m	n	nf	na
3	10	4	6
4	26	8	8
5	22	12	10
6	28	16	12
7	38	24	14

SOD can can also augment HTC factors with additional ETC factors (Examples 14-20). To use this facility, you have to enter a file which include the HTC factors in the File field, click Factors in the Add additional panel, and additional number of factors in the 2-level factor and 3-level factor fields.

Output

The output message containing the result of the best try will appear in a pop-up window when the program stops or when you stop the program by clicking the STOP button. It is also saved in the file sod.htm in the working directory. This file can be read by a browser such as IE, Firefox or Google Chrome. Information for this try includes:

1. Try number;
2. The random seed used;
3. The number of iterations;
4. The sum of the objective functions f+g, followed by f. If f=0 and Σx²_ix²_j=const., f will be followed by an (R) and the program stops. The constructed design is called slope-rotatable (Park, 1987).
5. det=|X'X| where X is the expanded design matrix;
6. det*=|X'X|^1/p/n where n is the number of runs and p is the number of parameters in the model.
7. Factor levels;
8. The non-zero sums of squares (in (1) and (2)) contributing to f+g.
9. The number of rows/columns which form the base design;
10. The time in seconds SOD used to construct the above design.

Example

1. A 3³ factorial design (http://designcomputing.net/gendex/sod/s27x3.html).
2. A 3⁴ factorial design (http://designcomputing.net/gendex/sod/s81x4.html).
3. An mixed-level design for 5 factors in 36 runs (http://designcomputing.net/gendex/sod/m36x5.html).
4. An BBD-type design for 3 factors in 15 runs (http://designcomputing.net/gendex/sod/b3.html).
5. An BBD-type design for 4 factors in 27 runs (http://designcomputing.net/gendex/sod/b4.html).
6. An BBD-type design for 5 factors in 46 runs (http://designcomputing.net/gendex/sod/b5.html).
7. An BBD-type design for 6 factors in 54 runs (http://designcomputing.net/gendex/sod/b6.html).
8. An BBD-type design for 7 factors in 62 runs (http://designcomputing.net/gendex/sod/b7.html).
9. A SCD for 3 factors in 10 runs (http://designcomputing.net/gendex/sod/c3.html).
10. A SCD for 4 factors in 16 runs (http://designcomputing.net/gendex/sod/c4.html).
11. A SCD for 5 factors in 21 runs (http://designcomputing.net/gendex/sod/c5.html).
12. A SCD for 6 factors in 28 runs (http://designcomputing.net/gendex/sod/c6.html).
13. A SCD for 7 factors in 38 runs (http://designcomputing.net/gendex/sod/c7.html).
14. A split-plot RSD with 1 HTC factor and 2 ETC factors in 4 blocks of 4 runs each (http://designcomputing.net/gendex/sod/pkv1.html).
15. A split-plot RSD with 1 HTC factor and 3 ETC factors in 4 blocks of 12 runs each (http://designcomputing.net/gendex/sod/pkv2.html).
16. A split-plot RSD with 1 HTC factor and 2 ETC factors in 3 blocks of 5 runs each (http://designcomputing.net/gendex/sod/pkv3.html).
17. A split-plot RSD with 2 HTC factors and 2 ETC factors in 9 blocks of 5 runs each (http://designcomputing.net/gendex/sod/pkv4.html).
18. A split-plot RSD with 1 HTC factor and 4 ETC factors in 3 blocks of 12 runs each (http://designcomputing.net/gendex/sod/pkv7.html).
19. A split-plot RSD with 1 HTC factor and 4 ETC factors in 3 blocks of 12 runs each (http://designcomputing.net/gendex/sod/pkv7bis.html).
20. A split-plot RSD with 2 HTC factors and 2 ETC factors in 10 blocks of 4 runs each (http://designcomputing.net/gendex/sod/ceramic.html).

Notes:

• Examples 1-2: These two are orthogonal designs.
• Examples 4-8: These designs correspond to those for 3-7 factors in Box & Behnken (1960) and Nguyen & Borkowski (2008). Note that the one for 6 factors improves the corresponding BBD in terms of optimality as well as rotatability (see Nguyen & Borkowski, 2008).
• Examples 9-13: These designs correspond to those for 3-7 factors in Draper & Lin (1990). Note that the one for 7 factor improves the corresponding SCD in terms of D-optimality.
• Examples 14-20: These split-plot RSDs were constructed by augmenting hard-to-change (HTC) factors with additional easy-to-change (ETC) factors. It can be verified that the constructed designs in these examples satisfy the the balanced equivalent estimation property expressed in equation (4) of Parker, et al. (2006). Designs in Examples 14-19 correspond to those in Table I-IV and VII and IX in Parker et al. (2006). The design in Example 20 corresponds to the one in Table 5 in Vining et al. (2005).

References

Box, G.E.P. & Behnken, D.W. (1960) Some new three level designs for the study of quantitative variables. Technometrics 2, 455-477.
Box, G.E.P. & Wilson, K.B. (1951) On the experimental attainment of optimum conditions. J. Roy. Statist. Soc. Ser. B 13, 1-45.
Draper, N.R. & D.K.J. Lin (1990) Small response surface designs. Technometrics 2, 187-194.
John, P.M.W. (1971) Statistical design and analysis of experiments, New York: McMillan.
Nguyen, N-K. (1996a) An algorithmic approach to constructing supersaturated designs. Technometrics 38, 205-209.
Nguyen, N-K. (1996b) A note on the construction of near-orthogonal arrays with mixed levels and economic run size.Technometrics 38, 279-283.
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 & D.K.J. Lin (2010) A note on small composite designs for sequential experimentation, Accepted for Journal of Statistics Theory and Practice.
Park, S. H. (1987) A class of multifactor designs for estimating the slope of response surfaces. Technometrics 29, 449-453.
Parker, A.P., Kowalski, S.M. & Vining, G.G. (2006) Classes of split-plot response surface designs for equivalent estimation. Quality & Reliability Engineering International 22, 291-305.
Vining, G.G, Kowalski, S.M. & Montgomery, D.C. (2005) Response surface within a split-plot structure. J. of Quality Technology 37, 115-129.

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