Some Results On2^{（n₁+n₂）-（k₁+k₂）}8¹Designs Containing Clear Effects

Fractional factorial （FF） designs are widely used for factorial experiments. There areseveral optimality criteria for selecting designs, such as the maximum resolution （MR）criterion （Box and Hunter,1961）, the minimum aberration （MA） criterion （Fries andHunter,1980）, the clear efects （CE） criterion （Wu and Chen,1992） and the generalminimum lower-order confounding （GMC） criterion （Zhang, Li, Zhao and Ai,2008）. Ifthe prior knowledge is known, the experimenters usually use the CE or GMC criterion.The mixed-level orthogonal table is commonly used in experimental designs. Whenlevels of some factors are not equal in quantity, we should select the mixed-level designs.This paper considers the designs with n two-level factors and m eight-level factors, whichare usually denoted by2ⁿ8^m. Such designs can be constructed from two-level designsby the method of replacement, which was first formally introduced by Addelman （1962）.Zhao and Chen （2011） gave the necessary and sufcient conditions for the designs with ntwo-level factors and1four-level factor containing clear efects.If the levels of some factors are difcult to be changed or controlled, it may beimpractical to perform the experimental runs of FF designs in a completely random order.This motivates us to use fractional factorial split-plot （FFSP） designs to meet the specialdemands. The FFSP designs contain two sections, the whole-plot （WP） and the sub-plot（SP） section. Factors in the two sections are called WP factors and SP factors respectively.Suppose that there are n factors in a design, then the n1hard-to-change factors are WPfactors, and the rest n2factors are SP factors （n1+n2=n）. Let k1and k2be the numberof defining words in WP section and SP section respectively, then such an FFSP designcan be denoted by2^{（n₁+n₂）-（k₁+k₂）}. If there are both two-level and eight-level factors in anexperiment and it is difcult to change or control the levels of some factors, a mixed-levelFFSP design, which can be denoted by2^{（n₁+n₂）-（k₁+k₂）}8^m, can be used.This paper considers the regular mixed-level FFSP designs2^{（n₁+n₂）-（k₁+k₂）}8¹. It con-sists of three chapters. Chapter1introduces the basic definitions related to the FF design,optimality criteria and the FFSP design. Chapter2gives a complete classification of the 2^{（n₁+n₂）-（k₁+k₂）}81designs containing clear efects. Section2.1gives a simple summary onthe literature. Section2.2introduces the notations and definitions of two types of mixed-level FFSP designs,2^{（n₁+n₂）-（k₁+k₂）}8_s¹and2^{（n₁+n₂）-（k₁+k₂）}8_w¹, according to the diferenceof the eight-level factor in WP section or SP section, and gives the concept of three typesof two factor interaction components. Sections2.3and2.4study the conditions for res-olution III and IV2^{（n₁+n₂）-（k₁+k₂）}8_s¹designs containing clear efects. Section2.5givesthe sufcient and necessary conditions for resolution III and IV2^{（n₁+n₂）-（k₁+k₂）}8_w¹designscontaining clear efects. Chapter3gives a brief concluding to the paper.