By considering the so-called algebraic orthogonality among experimental runs, this paper introduces a new criterion, minimum inner-product moment (MIPM), for general asymmetrical designs, and shows that MIPM is equivalent to the minimum moment aberration (MMA) criterion for the natural weights. Furthermore, the relationship between the generalized minimum aberration (GMA) and some model-dependent efficiency criteria is investigated by using the complex contrasts. Thus, two new justifications of GMA criterion is given from the points of view of orthogonality among experimental runs and design efficiency. They are generalizations of the related results of Butler (2003) and Cheng, Deng, and Tang (2002) for twolevel factorial designs.

In this paper, the minimum aberration criterion is extended for choosing blocked fractional factorial designs. Ideally, one should seek a design that has minimum aberration with respect to both treatments and blocks. We prove the nonexistence of such a design. For this reason, it is needed to compromise between the wordlength pattern of blocks and that of treatments. By exploring the wordlength patterns of a two-level fractional factorial design, we introduce a concept of alias pattern and give accurate formulas for calculating the number of alias relations for any pair of orders of treatment effects as well as of treatment and block e ects. According to the structure of alias pattern and the hierarchical principles on treatment and block e ects, a minimum aberration criterion for selecting blocked fractional factorial designs is studied. Some optimal blocked fractional factorial designs are given and comparisons with other approaches are made.

It is very powerful for constructing nearly saturated factorial designs to character-iza fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary worldength pattern of a regular symmetrical fractional factorial split-plot (FFSTP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial dwsign theory and coding theory, we obtain some general and unified combinato-rial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to theses identites, we further estab-lish some general and unified ruleds for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.

It is known by Zhang and Park (J. Statist. Plann. Inference 91 (2000) 107) that there are no minimum aberration (MA) designs with respect to both treatments and blocks for blocked regular mixed-level factorial designs. So it should be compromised between the block wordlength pattern and treatment wordlength pattern. Two methods are considered in this article. The first isMAblocking scheme of an MA design. The other is to combine the components of the two wordlength pattern vectors into one combined wordlength pattern according to the modified hierarchical assumptions and an appropriate ordering of the numbers of alias or confounding relations. The relationship between the two types of optimal blocked designs is investigated. A complete catalogue of optimal blocked regular mixed factorial designs of the above two types with 16 or 32 runs is given.

Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considerign the blocking factor. In this pape, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocded combined-optimal designs. Some general results are obtained which relate 2k-pIII inital designs with their complementary designs when k>5n8, where n=2k-p. By applying these results, we are able to characterize 2k-pIII combined-optimal designs or blocked combined-optimal dwdigns in terms of their complementary designs. It is also proved that both 2k-pIII compbined-optimal and blocked combined-optimal designs are not minimum aberration designs when k>5n8 and n-1-k>2. And some combined-optimal and bloked combined -optimal designs with 16 and 32 runs are constructed fo rillustration.