## 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 one at 2-level) in 26 runs (including two center runs). Also, you want each 3-level factor to have 8 zeros. At the working directory, type the following command at the command prompt (case is important and no space is allowed before/after the equal sign):

java -cp c:\gendex sod m2=3 m3=1 n=16 ncenters=2 nzeros=8

**sod** will start running and after try 1,
the plan of the constructed design for this try will be displayed at the terminal window
(as all *A*'s have reached 0) and then **sod** stops:

sod 10.0: Constructing 3-level and Mixed-level Second-Order Designs (c) 2018 Design Computing (http://designcomputing.net/) Note: Best design for 4 factors in 18 runs. Try iter A1 A2 rmax rave trace det* 1 9 0 0 0 0 1.2 0.64 Design Plan for try 1 using seed 1533555013655: 1 1 1 -1 -1 -1 1 1 0 1 -1 -1 1 -1 1 1 0 1 -1 -1 1 1 -1 1 1 -1 -1 -1 0 -1 -1 1 0 -1 1 -1 -1 1 -1 1 -1 1 1 -1 0 -1 -1 1 0 -1 1 -1 0 1 1 1 -1 -1 -1 -1 0 1 1 1 0 0 0 0 0 0 0 0 X'X 18 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 8 0 0 0 16 0 0 16 0 16 inverse(X'X) 0.1 -0.1 0 0 0 0 0 0 0 0 0 -0 0.225 0 0 0 0 0 0 0 0 0 -0 0.125 0 0 0 0 0 0 0 0 -0 0.062 0 0 0 0 0 0 0 -0 0.062 0 0 0 0 0 0 -0 0.062 0 0 0 0 0 -0 0.125 0 0 0 0 -0 0.125 0 0 0 -0 0.125 0 0 -0 0.062 0 -0 0.062 -0 0.062 Note: sod used 0.171 seconds. Note: sod.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 sod m2=3 m3=1 n=16 ncenters=2 nzeros=8 seed=1234 tries=10000

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:

_{c }m |
n |
n_{f} |
n_{c} |
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 and the number of 2-level factorial runs using the preset values

_{a}*n*and

_{a }*n*in Table 2.Here is an example:

_{f}java -cp c:\gendex sod in=a3.txt n=4

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 at the terminal 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

- 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).
- 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 2-5: These designs correspond to those for 3-5 factors in Box & Behnken (1960) and Nguyen & Borkowski (2008).
- Examples 6-10: 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.

©2000-2018 Design Computing