Optimal factor assignment for asymmetrical fractional factorial designs: Theory and applications.

Abstract

Fractional factorial designs have been successfully used in various scientific investigations for many decades. Its practical success is due to its efficient use of experimental runs to study many factors simultaneously. A fundamental and practically important question for factorial designs is the issue of optimal factor assignment to columns of the design matrix. Aimed at solving this problem, this thesis introduces two new criteria: the generalized minimum aberration and the minimum moment aberration, which are extensions of the minimum aberration and minimum <italic>G</italic>₂-aberration. These new criteria work for symmetrical and asymmetrical designs, regular and nonregular designs, orthogonal and nonorthogonal designs, nonsaturated and supersaturated designs. They are equivalent for symmetrical designs and in a weak sense for asymmetrical designs. The theory developed for these new criteria covers many existing theoretical results as special cases. In particular, a general complementary design theory is developed for asymmetrical designs and some general optimality results for mixed-level supersaturated designs. As an application, a two-step approach is proposed for finding optimal designs and some 16-, 18-, 27- and 36-run optimal designs are tabulated. As another application, an algorithm is developed for constructing mixed-level orthogonal and nearly orthogonal arrays, which can efficiently construct a variety of small-run arrays with good statistical properties.