Factorial experiments are widely used in industry to investigate the effects of process factors on quality response variables. Many food processes, for example, are not only subject to variation between days, but also between different times of the day. Removing this variation using blocking factors leads to rowcolumn designs. In this paper, an algorithm is described for constructing factorial row-column designs when the factors are quantitative, and the data are to be analysed by fitting a polynomial model. The row-column designs are constructed using an iterative interchange search, where interchanges that result in an improvement in the weighted mean of the efficiency factors corresponding to the parameters of interest are accepted. Some examples illustrating the performance of the algorithm are given.
Key Words: Design of Experiments, Optimal Design, Simulated Annealing, Trends.
By STEVEN G. GILMOUR Queen Mary, University of London, London E1 4NS, UK LUZIA A. TRINCA CP 510, UNESP, 18618-000 Botucatu, SP, Brazil
RESPONSE surface methodology (RSM) is a set of techniques for empirically studying processes by designing experiments, modelling the resulting data, and interpreting the .tted models. At least initially, low order polynomial models are usually fitted, often the second order model,
where y is the response variable, x1, ..., xq are the levels of the continuous factors being varied, coded to be between -1 and 1, and is the error term. For a full discussion of RSM, see the books by Box and Draper (1987), Myers and Montgomery (1995), and Khuri and Cornell (1996). For more recent developments, see Myers (1999).
Traditionally, response surface (RS) designs have been described as if the treatments are completely randomised to the experimental units, but recently there has been interest in arranging these designs in other unit structures. Atkinson and Donev (1989), Cook and Nachtsheim (1989), and Trinca and Gilmour (2000) gave algorithms for arranging RS designs in blocks. Trinca and Gilmour (2001) gave an algorithm for arranging RS designs in general nested unit structures, such as split plots. Edmondson (1991) showed how four-level RS designs, based on two-level pseudo-factors, could be arranged in various unit structures using confounding of the appropriate e.ects in the pseudo-factors. Until now, there has been no discussion of how to arrange RS designs in general row-column or other crossed unit structures. This is the subject of the current paper. In the second section, we discuss food processing and other applications. Next, our criterion and algorithm are described. The algorithm is applied in the fourth section to three examples, data analysis is discussed in the fifth section, and in the sixth section some practical issues and extensions are discussed.
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