Dominic F. Vecchia

*Statistical Engineering Division, CAML*

Hari K. Iyer

C. T. Liao

*Colorado State University*

During the initial stages of a product or a process design, engineers
typically consider several factors which may influence on a performance
measure of interest. To understand the relative importance of each factor,
it is often desirable to run one or more screening experiments.
Traditionally, 2^{n-k} fractional factorial designs of resolution III, IV,
or V have been used for this purpose. Sometimes, however, it is possible
to obtain orthogonal designs with fewer runs than the traditional designs
by searching in the class of parallel-flats designs. Because such
designs need not have numbers of runs be a power of two, they may offer
considerable savings in time and expense over the usual fractional factorials.

We have developed and implemented an algorithm for constructing orthogonal
parallel-flats designs to meet user specifications. Specifically,
we suppose an investigator can partition the full set of factorial effects
into three disjoint sets:

- 1.
- Primary effects
*G*_{1}: those for which unbiased estimates are required
- 2.
- Secondary effects
*G*_{2}: those for which unbiased estimates are not
required at this stage, but which may be nonnegligible
- 3.
- Negligible effects
*G*_{3}: those believed to be negligible.

The objective is to find designs suitable for estimating all effects in
*G*_{1} based on a factorial linear model in which the effects in *G*_{3} are
assumed to be zero. Any such design is called a design of resolution
(*G*_{1}, *G*_{2}). Commercial software for this problem is based on an
exhaustive search for a suitable plan among single-flat designs.

Our algorithm is based an expression for the general element of the
information matrix
of an arbitrary parallel-flats design,
where
is the design matrix in the linear model
.
Although the algorithm is not guaranteed to find the minimum-run design
for a given problem, in nearly all of the tests conducted so far it has
produced an orthogonal design with run size equal to or smaller than various
published designs for estimating the same set of factorial effects.

To test the algorithm, we created several nonisomorphic sets *G*_{1} of
randomly selected primary effects with as many as 20 factors.
(*G*_{2} was taken to be the empty set for this exercise.)
In each case we included all main effects, a specified number of two-factor
interactions, and a specified number of three-factor interactions.
Each interaction was forced to include at least one of a specified set
of 1, 2, 3, or 4 "required'' factors.

The table shows the success rate in finding a design smaller than smallest
possible design that could be produced by traditional search algorithms
(e.g., 48 runs instead of 64; 80 or 96 runs instead of 128).
In the table, *n* is the number of factors, *x*_{2} is the number of
two-factor interactions in *G*_{1}, *x*_{3} is the number of three-factor
interactions in *G*_{1}, *r* is the number of required factors, at least
one of which must appear in every interaction, and the fraction *p*=*a*/*b*
shows the number *a* of *N*-run designs found in *b* trials.
(For some problems, 100 nonisomorphic sets *G*_{1} do not exist.)

*n* |
*x*_{2} |
*x*_{3} |
*r* |
*p* |
*N* |

12 |
20 |
0 |
2 |
1/1 |
48 |

12 |
20 |
0 |
3 |
100/100 |
48 |

12 |
20 |
0 |
4 |
1/100 |
48 |

12 |
18 |
2 |
2 |
35/100 |
48 |

12 |
18 |
2 |
3 |
1/100 |
48 |

12 |
18 |
2 |
4 |
0/100 |
48 |

16 |
16 |
0 |
2 |
48/49 |
48 |

16 |
16 |
0 |
3 |
39/100 |
48 |

16 |
16 |
0 |
4 |
7/100 |
48 |

16 |
15 |
1 |
2 |
18/100 |
48 |

16 |
15 |
1 |
3 |
8/100 |
48 |

16 |
15 |
1 |
4 |
0/100 |
48 |

16 |
14 |
2 |
2 |
11/100 |
48 |

16 |
14 |
2 |
3 |
2/100 |
48 |

16 |
14 |
2 |
4 |
1/100 |
48 |

*n* |
*x*_{2} |
*x*_{3} |
*r* |
*p* |
*N* |

16 |
20 |
0 |
2 |
20/28 |
48 |

16 |
20 |
0 |
3 |
6/100 |
48 |

16 |
20 |
0 |
4 |
0/100 |
48 |

16 |
19 |
1 |
2 |
7/100 |
48 |

16 |
19 |
1 |
3 |
1/100 |
48 |

16 |
19 |
1 |
4 |
1/100 |
48 |

18 |
14 |
0 |
1 |
1/1 |
48 |

18 |
14 |
0 |
2 |
50/51 |
48 |

18 |
14 |
0 |
3 |
24/100 |
48 |

18 |
14 |
0 |
4 |
4/100 |
48 |

20 |
12 |
0 |
2 |
41/41 |
48 |

20 |
12 |
0 |
3 |
46/100 |
48 |

20 |
12 |
0 |
4 |
0/100 |
48 |

18 |
46 |
0 |
3 |
3/3 |
80 |

18 |
46 |
0 |
4 |
26/100 |
96 |

In cases where the success rate appears to be very low (e.g., 1 out of 100),
it may be the case that 48-run designs do not exist for most of the trial
problems. The algorithm will find a subset of the ones that
actually exist, but the number of cases for which a 48-run design exists
is unknown and is generally very difficult to determine. Nevertheless,
the results in the table indicate that the algorithm can be expected to
be reasonably successful in finding 48-run designs of user-specified
resolutions (also some 80-run and 96-run solutions).