## Introduction

SOD is a program for constructing 3-level second-order response surface designs (RSDs) for
*m* factors in *n* runs
*D _{m}_{xn}* where each factor takes only
levels -1, 0 and +1. The constructed designs include Box-Behnken designs (BBDs) of Box &
Behnken (1960), Central Composite designs (CCDs) of Box & Wilson (1951),
Small Composite Designs (SCDs) of Draper & Lin (1990), etc. SOD can also augment a design 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 ∑*x _{i}*=∑

*x*=0 and ∑

^{3}_{i}*x*=∑

^{2}_{i}*x*=

^{4}_{i}*b*

_{i}where

*b*is the number of ±1 of factor

_{i}*i*. Let's impose the following conditions on

*D*:

*x*=0 for

^{2}_{i}x_{j}*i*<

*j*<

*k*;

(ii) ∑

*x*=0 for

^{2}_{i}x_{j}x_{k}*i*<

*j*<

*k*;

(iii) ∑

*x*=0 (and

_{i}x_{j}*x*=0) for

^{3}_{i}x_{j}*i*<

*j*;

(iv) ∑

*x*=0 for

_{i}x_{j}x_{k}*i*<

*j*<

*k*;

(v) ∑

*x*=0 for

_{i}x_{j}x_{k}x_{l}*i*<

*j*<

*k*<

*l*;

(vi) ∑

*x*-

^{2}_{i}x^{2}_{j}*b*/

_{i}b_{j}*n=0*for

*i*<

*j*.

where the summations are taken over the *n* design points . It
can be seen that these conditions are the conditions for *D* to be orthogonal (see Section 10.2
of John, 1971). A 3^{k} full factorial, an orthogonal design will satisfy all six conditions. The
CCDs and BBDs will satisfy the first five conditions (i)-(v), while the first three conditions
(i)-(iii) will imply the orthogonal quadratic effects(OQE) property (see Section 1 of Nguyen & Lin, 2011).

Let *A*_{1} be the sum of squares of the sums in (i)-(iii),
*A*_{2} be the sum of squares of the sums in (i)-(v) and
*A*_{3} be the sum of squares of the sums in (i)-(vi).
SOD construct *D* by starting with a 3-level design and
assign an equal number of ±1's to columns of
*D _{mxn}*. SOD then sequentially minimizes the

*A*'s by switching the positions of -1, 0 and +1 in each columns of

*D*. The detailed account of the SOD algorithm appears in Nguyen & Lin (2011). 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 GUI will pop up. Enter the number of factors at 2-level and 3-level, the number of factorial runs and center runs as shown.

Now, click **START**, the following window will pop up:

Choose 12 (=36/3) and click **OK**, SOD will start running and after try 1,
the plan of the constructed design for this try pops up in the SOD output window
(as all *A*'s have reached 0) and then SOD stops:

The **START** button has changed to **RESET** button. If you
click this **RESET** button, the output in the SOD window will
disappear and you can use SOD for a new design problem. 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.

To construct the BBD-type designs for 3-7 factors (Examples 4-8) you have to
use the preset values *n _{f}* (factorial runs from a
3

^{m}factorial),

*n*(center runs) and 0's (number of 0-level for each 3-level factor) in Table 1:

_{0 }m |
n |
n_{f} |
n_{0} |
0's |
---|---|---|---|---|

3 | 15 | 12 | 3 | 4 |

4 | 27 | 24 | 3 | 12 |

5 | 46 | 40 | 6 | 24 |

6 | 54 | 48 | 6 |
24 |

7 | 62 | 56 | 6 | 32 |

To construct the CCDs of Box & Wilson (1951) and the SCDs of Draper & Lin (1990) in examples 9-13, you have to specify a file which include the

*n*axial runs in the

_{a}**File**field and the number of factorial runs in the

**Factorial runs**field using the f ollowing preset values

*n*and

_{a }*n*in Table 2:

_{f}m |
n |
n_{f} |
n_{a} |
---|---|---|---|

3 | 10 | 4 | 6 |

4 | 16 | 8 | 8 |

5 | 22 | 12 | 10 |

6 | 28 | 16 | 12 |

7 | 38 | 24 | 14 |

## Output

The result of the best try is displayed in the SOD output window and is also saved in the file
SOD.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;
- The value of
*A*'s. If*A*_{2}=0 and ∑*x*=^{2}_{i}x^{2}_{j}*const.*,*A*_{3}will be followed by an (R) indicating that the constructed design is slope-rotatable (see Park, 1987). The program automatically stops when*A*_{3}reaches 0 or when the constructed design is slope-rotatable. - det*=|
**X'X**|^{1/}/^{p}*n*where*n*is the number of runs and*p*is the number of parameters in the model. - det=|
**X'X**| where**X**is the expanded design matrix; - The design plan and the associated random seed;
- The non-zero elements in
*J*. - The number of runs of the base design if any;
- The time in seconds SOD used to construct the above design.

## Example

- A 3
^{3}factorial design (http://designcomputing.net/gendex/sod/s1.html). - A 3
^{4}factorial design (http://designcomputing.net/gendex/sod/s2.html). - An RSD for 4 factors in 27 runs (http://designcomputing.net/gendex/sod/s3.html).
- An mixed-level RSD for 5 factors in 36 runs (http://designcomputing.net/gendex/sod/s4.html).
- An BBD-type design for 3 factors in 15 runs (http://designcomputing.net/gendex/sod/b3.html).
- An BBD-type design for 4 factors in 27 runs (http://designcomputing.net/gendex/sod/b4.html).
- An BBD-type design for 5 factors in 46 runs (http://designcomputing.net/gendex/sod/b5.html).
- An BBD-type design for 6 factors in 54 runs (http://designcomputing.net/gendex/sod/b6.html).
- An BBD-type design for 7 factors in 62 runs (http://designcomputing.net/gendex/sod/b7.html).
- A SCD for 3 factors in 10 runs (http://designcomputing.net/gendex/sod/c3.html).
- A SCD for 4 factors in 16 runs (http://designcomputing.net/gendex/sod/c4.html).
- A SCD for 5 factors in 21 runs (http://designcomputing.net/gendex/sod/c5.html).
- A SCD for 6 factors in 28 runs (http://designcomputing.net/gendex/sod/c6.html).
- A SCD for 7 factors in 38 runs (http://designcomputing.net/gendex/sod/c7.html).

**Notes**:

- Examples 1-2: These two are orthogonal designs.
- Examples 5-9: 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 10-14: These designs with OQE property 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.

## 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 (2011) A note on small composite designs for
sequential experimentation, *Journal of Statistics Theory and
Practice 5 109-117*.

Park, S. H. (1987) A class of multifactor designs for estimating the slope of response surfaces.

*Technometrics*

**29**, 449-453.

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