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3:5
August 31, 2006
Char Count= 0
JWBK119-17
Infeasible Conditions 259
Table 17.1 2 4−1 design with 4 = 123.
I X 1 X 2 X 3 X 4
1 − − − −
2 + − − +
3 − + − +
4 + + − −
5 − − + +
6 + − + −
7 − + + −
8 + + + +

where certain specific combinations are infeasible or best avoided, for reasons already
explained. Under such circumstances, it would be useful to consider the possible vari-
ants of fractional factorials for actual use. For example, if four factors are to be studied
in eight experimental runs, the recommended design is usually 2 4−1 with 4 = 23 as
the generator, resulting in the matrix shown in Table 17.1. However, if run no. 8, say, is
physically undesirable because it requires an ‘all high (+)’setting for the factors, then
some alternative designs can be considered, such as one shown in Table 17.2 which is
based on 4 =−123. If, on further planning, it is felt that one particular combination
of factor settings, say no. 7 in Table 17.2, is operationally difficult to arrange, then yet
other designs can be used to avoid the (−, +, +, +) requirement; one such design is
shown in Table 17.3, which is based on 4 =−23.
In some other situations, data may already exist for a particular factor setting combi-
nation, so it would be desirable to include that combination in the design. For example,
if (+, −, +, +) is to be included, whereas (+, +, −, +) is to be excluded, then a ma-
trix generated by 4 =−12 would be appropriate. In all, as many as eight alternative
2 4−1 designs are available for the study of four factors in eight experimental runs,
based on generators 4 = 123, 4 =−123, 4 = 12, 4 =−12, 4 = 13, 4 =−13, 4 = 23 and
4 =−23. The structures of design matrices based on these generators are summa-
rized under columns A, B, C, D, E, F, G and H respectively in Table 17.4, showing

Table 17.2 2 4−1 design with 4 =−123.
I X 1 X 2 X 3 X 4

1 − − − +
2 + − − −
3 − + − −
4 + + − +
5 − − + −
6 + − + +
7 − + + +
8 + + + −