21.4.3.9. Small Composite Design (SCD)
Hartley (1959) pointed out that, for estimation of the second-order surface, the cube portion of the central composite design (CCD) need not be of resolution \(V\). It could be of resolution as low as \(III\), provided that two-factor interactions were not aliased with other two-factor interactions. Hartley employed a smaller fraction of the \({{2}^{k}}\) factorial than is used in the original Box-Wilson designs and so reduced the total number of points. Hartley’s cubes may be designated resolution \(II{{I}^{*}}\), meaning a design of resolution \(III\) but with no words of length four in the defining relation.
Westlake (1965) provided a method for generating composite designs based on irregular fractions of the \({{2}^{k}}\) factorial system rather than using the complete factorials or regular fractions of factorials employed by Box and Wilson (1951) and Hartley (1959). Westlake provided three examples for 22-run designs for \(k=5\), one example of a 40-run design for \(k=7\), and one example of a 62-run design for \(k=9\).
Draper (1985, 1990) proposed an alternative approach to obtaining small composite design (SCD), which employed columns of the Plackett-Burman designs (1946) rather than regular or irregular fractions. Draper (1985) and Draper and Lin (1990) have shown that many small composite designs exist. The formation of these designs is constructed by (1) using the \(2k\) axial runs plus center runs, (2) adding the \(k\) columns of a Plackett-Burman design for the cube portion to avoid singularity or near singularity, (3) while removing one of each set of duplicates if duplicate runs exist.
Draper provided, using 12-run, 28-run and 44-run Plackett-Burman designs, 22-run design for \(k=5\), 42-run design for \(k=7\) and 62-run design for \(k=9\), respectively. However, his approach cannot give a general design assessed on the number of factors, because it is another optimization problem. For the detailed information, one may refer to the references (Draper, 1985; Draper and Lin, 1990).
In AutoDesign, an automated version of SCD is presented. This design gives slightly super-saturated samplings for the second-order response surface model. We call it as a generalized small composite design (GSCD).
For the large scaled second-order response surface model, GSCD still requires many samplings, even though it can reduce the number of sampling points than CCD and BBD. Hence, we develop new sub-saturated sampling methods. We call them as incomplete small composite designs (ISCD). They are divided into two methods. One is an incomplete small composite design-I (ISCD-I). Another is an incomplete small composite design-II (ISCD-II). The ISCD-I reduces the points in the cube portion of GSCD. Then, the ISCD-II removes the star points of the ISCD-I.
Figure 21.142 Geometric views of three small composite designs
Table 21.9, the total number of points in cube and star, excluding center point, in various small composite designs previously discussed, are summarized.
Factors, k |
|||||||||
|---|---|---|---|---|---|---|---|---|---|
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
||
Number of unknown coefficients |
10 |
15 |
21 |
28 |
36 |
45 |
55 |
66 |
|
Number of stat points |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
20 |
|
Points for cube portion |
Box and Hunter(1957) |
8 |
16 |
16 |
32 |
64 |
64 |
128 |
128 |
Hartly(1959) |
4 |
8 |
- |
16 |
32 |
- |
64 |
- |
|
Westlake(1965) |
- |
- |
12 |
- |
26 |
- |
44 |
- |
|
Draper(1985) |
- |
- |
12 |
- |
28 |
- |
44 |
- |
|
Draper and Lin(1990) |
- |
- |
11 |
- |
22 |
- |
38 |
- |
|
Proposed GSCD |
4 |
8 |
8 |
20 |
24 |
32 |
44 |
48 |
|
Proposed ISCD-I |
4 |
8 |
8 |
8 |
8 |
12 |
12 |
12 |
|
The symbol ‘-’ denotes that the design is not provided by the authors.
Reference
Box, C.E.P. and Hunter, W.G., 1957, “Multi-factor Experimental Designs for Exploring Response Surfaces”, Annals of Mathematical Statistics, Vol. 28, pp. 195~241.
Hartly, H.O., 1959, “Small Composite Design for Quadratic Response Surfaces”, Biometrics, Vol. 15, pp.611~624.
Westlake,W.J., 1965, “Composite Design based on Irregular Fractions of Factorials”, Biometrics, Vol. 21, pp. 324~336.
Draper, N.R., 1985, “Small Composite Designs”, Technometrics, Vol. 27, No. 2, pp. 173~180.
Draper, N.R. and Lin, D.K., 1990, “Small Response Surface Design”, Technometrics, Vol. 32, No. 2, pp. 187~194.
Kim M.-S and Heo S.-J., 2003, “Conservative Quadratic RSM combined with Incomplete Small Composite Design and Conservative Least Squares Fitting”, KSME International Journal, Vol. 17, No. 5, pp. 698~707.