## 3.3.7 Orthogonal Designs of User-Specified Resolution

Dominic F.  Vecchia

Statistical Engineering Division, CAML

Hari K. Iyer

C. T. Liao

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, 2n-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 G1: those for which unbiased estimates are required
2.
Secondary effects G2: those for which unbiased estimates are not required at this stage, but which may be nonnegligible
3.
Negligible effects G3: those believed to be negligible.

The objective is to find designs suitable for estimating all effects in G1 based on a factorial linear model in which the effects in G3 are assumed to be zero. Any such design is called a design of resolution (G1, G2). 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 G1 of randomly selected primary effects with as many as 20 factors. (G2 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, x2 is the number of two-factor interactions in G1, x3 is the number of three-factor interactions in G1, 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 G1 do not exist.)

 n x2 x3 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 x2 x3 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).

Date created: 7/20/2001
Last updated: 7/20/2001