Multi-factorial design

Apart from new catalytic methods, cascade conversions require new process technologies, such as in situ product recovery, reactor design, and compartmental-ization. In the long term, part of the present-day stoichiometric chemistry as well as bio- and chemocatalytic conversions in multi-step syntheses will gradually be replaced by cascade catalysis in concert, and full fermentations by cell factory design, or combinations thereof (Fig. 13.17). 

Complete multi-level factorial designs would usually yield too large a number of test systems for a first approach to new reaction systems. It is possible to reduce the number of test systems and yet achieve a selection which covers a large part of the entire reaction space. This can be achieved by a selection made from a two-level fractional factorial design. The principles are illustrated by an example provided by the Willgerodt-Kindler reaction. 

The arrangement of experimental runs in a factorial design makes it possible to use each observed response more than once. Each result is actualfy used in combinations with the other results to compute an average response, all main effects, all two-factor interaction effects,... all multi-factor interaction effects. This means that each individual experimental run is equalfy important to the overaU result. 

In the previous chapters it was seen that the linear coefficients, Bj, and the rectangular (interaction) coefficients. By, can be efficiently estimated by a two-level factorial or fractional factorial design. To determine also the square coefficients, Bjj, it will, however, be necessary to explore the variations of the experimental variables on more than two levels. One possibility would be to use a multi-level factorial design to define a grid of experimental points in the domain. However, with r levels and k variables, the number of experiments, increases rapidly and becomes prohibitively large when the number of levels and the number of variables increase. 

In the section below, three examples are given of how the principles of factorial and fractional factorial designs can be applied in the selection of test systems. In the next chapter, an example is given of how a multi-level factorial design in the principal properties was used in conjunction with PLS modelling to analyze which properties of the reaction system are responsible for controlling the selectivity in the Fischer indole reaction. 

Another factorial design, used for studying solubility in mixed micelles, introduces and demonstrates multi-linear regression and analysis of variance. It is then extended, also in chapter 5, to a central composite design to illustrate the estimation of predictive models and their validation. 

The coefficients in the model equation 3.4 may be estimated as before, as linear combinations or contrasts of the experimental results, taking the columns of the effects matrix as described in section III.A.5 of chapter 2. Alternatively, they may be estimated by multi-linear regression (see chapter 4). The latter method is more usual, but in the case of factorial designs both methods are mathematically equivalent. 

A. A. Momen, G. A. Zachariadis, A. N. Anthemidis and J. A. Stratis, Use of fractional factorial design for optimisation of digestion procedures followed by multi-element determination of essential and non-essential elements in nuts using ICP-OES technique, Talanta, 2007,71(1), 443-451. 

For the experiment array, I prefer an orthogonal central-composite design (2), (3), which consists of three main parts, as shown in Table I. The first is a conventional 16-experiment fractional factorial design for five variables at two levels. The second comprises three identical experiments at the average, or center-point, conditions for the first 16 experiments. The final part comprises two out-lier experiments for each variable. These augment the basic two level design to provide an estimate of curvature for the response to each variable. The overall effect of the design is to saturate effectively the multi-dimensional variable space. It is more effective than the conventional "one-variable-at-a-time" approa.ch. 

Considering only production network design, Schmenner (1979) builds on the focused factory concept to develop four distinct multi-plant strategies. While he does not consider an international environment, the generic strategies developed for domestic networks are also applied to international production networks (cf. Kouvelis et al. 2004, p. 127). Based on a product/market or process focus Schmenner defines four plant types ... 

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