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Performing successful experiments is a crucial topic for scientists. The selection of factors and levels that have significant effects on the outputs is the most hard problem they may face. For a perfect procedure, a small number of runs (trials) are preferred as an initial-stage experiment (ISExp) for gaining prior information about the influence (importance) of the suggested factors and levels. After analysing the ISExp, the following prior information may arise: (i) the number of parameters in the used model is much larger than the expected number of runs and thus more runs need to be added; (ii) some of the ignored or fixed factors may be active and need further investigations; (iii) extending the levels of some active factors is necessary for optimising the outputs; and/or (iv) the impacts of the active factors are not equally important and thus different weights need be assigned to the factors to identify their importance. Therefore, follow-up experiments (FuExps) need to be conducted for adding new active factors, levels, runs, and/or factor-weights to the ISExps. This paper provides novel techniques for designing FuExps to deal with these real-life scenarios for ISExp with a mixture of two-level factors and four-level factors. To clarify the power and efficiency of the proposed techniques, numerical and theoretical justifications are given. [ABSTRACT FROM AUTHOR]

The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the experimenters do not know which factors turn out to be important before conducting the experiment. Therefore, designs with good projections are desirable for factor screening. In this regard, significant questions are arising such as whether the optimal full-dimensional designs have good projections onto low dimensions? How experimenters can measure the goodness of a full-dimensional design by focusing on all of its projections?, and are there linkages between the optimality of a full-dimensional design and the optimality of its projections? Through theoretical justifications, this paper tries to provide answers to these interesting questions by investigating the construction of optimal (average) projection designs for screening either nominal or quantitative factors. The main results show that: based on the aberration and orthogonality criteria the full-dimensional design is optimal if and only if it is optimal projection design; the full-dimensional design is optimal via the aberration and orthogonality if and only if it is uniform projection design; there is no guarantee that a uniform full-dimensional design is optimal projection design via any criterion; the projection design is optimal via the aberration, orthogonality and uniformity criteria if it is optimal via any criterion of them; and the saturated orthogonal designs have the same average projection performance. [ABSTRACT FROM AUTHOR]