Theory of Orthogonal Designs

Speeding up the Search

For designs with many factors or blocks, the tree of candidate confounding rules can be very large and the search can take a very long time. In these cases, the FACTEX procedure spends a lot of time exploring sets of rules that are essentially the same and that all result in failure. A technique for pruning the search tree (see Figure 16.1) is as follows. Suppose that for some selection r_i for rule i, all the branches for the next rule eventually result in failure. Then any other selection r'_i is immediately declared a failure if the resulting number of candidates is the same as for the failed rule r_i. The search goes on to the next selection for rule i.

This method of pruning is not perfect; it may prune a branch of the search tree that would have resulted in a success. In mathematical terms, candidate sets C_i are not necessarily isomorphic because they have the same size. You can use the NOCHECK option in the PROC FACTEX statement to turn off the pruning. With the NOCHECK option, the FACTEX procedure searches the entire tree of feasible confounding rules; and if given enough time, will find a design if one exists. The default argument for the TIME= option on the PROC FACTEX statement limits the search time to one minute.

Figure 16.1: Search Tree

On the other hand, you should recognize how rarely the NOCHECK option is needed to produce a design with a given resolution. For example, consider all possible blocked and unblocked two-level designs with minimum resolution for 50 or fewer factors and 128 or fewer runs. Of the 849 different designs, the NOCHECK option is required in only five cases. The five designs for which the NOCHECK option is required are listed in Table 16.1. Note that all of these are block designs, most for many factors and relatively small blocks.

Table 16.1: Designs Requiring the NOCHECK Option

Number of	Number of	Block
Factors	Runs	Size	Resolution
5	16	2	4
21	32	4	3
22	32	4	3
23	32	4	3
39	64	4	3

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