Optimization one-factor-at-a-time

A one-factor-at-a-time optimization is consistent with a commonly held belief that to determine the influence of one factor it is necessary to hold constant all other factors. This is an effective, although not necessarily an efficient, experimental design when the factors are independent. Two factors are considered independent when changing the level of one factor does not influence the effect of changing the other factor s level. Table 14.1 provides an example of two independent factors. When factor B is held at level Bi, changing factor A from level Ai to level A2 increases the response from 40 to 80 thus, the change in response, AR, is... 

TABLE 9.3. Experiment design for one-factor-at-a-time optimization (two-factor five-level). 

Secondly, with the OVAT approach the importance of interactions is not taken into account. An interaction between two factors is present when the effect of one factor depends on the level of another factor. Since only one factor at a time is varied, the presence or absence of interactions cannot be verified. However, this is not dramatic, since in robustness testing the interaction effects are considered negligible. The evaluation of such interactions is more important in method optimization. 

Fig. 11. The response surface of a two-factor system. The lines represent equi-response lines. Optimization by varying one factor at a time. From P. J. Golden and S. N. Deming, Laboratory Microcomputer 3, 44 (1984). Reproduced by permission of Science Technology Letters, England...

As mentioned before, people tend intuitively to turn to the one-variable-at-a-time technique for its conceptual simplicity, and ignore the possible interaction between independent variables. A good example of the interaction between factors is that between enzyme concentration (E) and reaction temperature (T). Assuming E and T are the chosen factors for optimization, one possible interaction will be that T tends to influence the way E affects the conversion yield and vice versa. Since reaction temperature increased, enzyme activity was suppressed than at low temperature and the rate of enzyme-catalysis is affected by temperature this will inevitably affect conversion yield of the product. Should the interaction be minor or negligible, a one-factor-at-a-time search will give a satisfactory result. 

An experiment design for one-variable-at-a-time optimization is shown in Table 9.3. The experiment involves five levels for each factor, expressed in coded form which can be linearly transformed back to corresponding true values, so that the arrangement could be applied to other systems. In the coded form, one unit could represent 10 °C difference in reaction temperature,... 

DOE is an alternative to best-guess or one-factor-at-a-time experiments, which are time- and resource-intensive, and may not produce the optimal solution in the end. By using DOE to test more than one factor at a time, you ll end up with better, more reproducible solutions in less time, and you ll expend fewer resources. However, the approach does require rigorous statistical analysis and should only be used with support from statisticians or others who have been trained in DOE. 

Knowledge of multivariate methods is not, however, widely spread in the community of synthesis chemists. Therefore, many new methods are still being investigated through poorly designed experiments and hence, new procedures are not properly optimized. Still, the most common method to carry out "systematic studies" is to consider "one factor at a time", although such an approach was shown by R.A. Fisher to be inappropriate over 60 years ago [1], when several factors are to be considered. 

FIGURE 2.2. One-variable-at-a-time optimization procedure for two factors, Xi and X2, in the presence of an interaction effect between the factors. Dotted lines = hypothetical contour plot of response to optimize. A = starting point B = best result after varying xi a first time C = best result after varying X2 a first time (= usually reported optimum) and D = best result after varying xi a second time (= real optimum). 

OPTIMIZATION THROUGH THE ONE-FACTOR-AT-A-TIME (OFAT) EXPERIMENTATION 515... 

Optimization through the one-factor-at-a-time (OFAT) experimentation... 

EPI has its maximal value of unity for well-watered conditions, optimal temperatures for net CO2 uptake, and saturating PAR — under other conditions, EPI represents the fraction of maximal net CO2 uptake expected. To calculate EPI for some species in the field, its net CO2 uptake over 24-h periods should be measured in the laboratory while varying one factor at a time, and the environmental conditions in the field should be specified, preferably on a daily basis. Models can be used to predict certain field conditions, such as soil water status as a function of rainfall and time, and monthly averages can sometimes be used, such as for calculating the Temperature Index (3). 

An appropriate experimental design (called a factorial design) can give one this information, while a one-factor-at-a-time approach never can. Thus we see that DOE is of great importance in studies such as the one on optimizing situations in chemical process yield. We now consider the subject of DOE in greater detail. 

In the optimization of tablet formulations, different approaches can be used. The one variable at a time method requires many experiments and there is no guarantee that an optimal formulation is achieved. Moreover the interaction between different factors, which may influence the tablet properties, will not be detected [10]. The use of an experimental design can be helpful in the optimization of tablet formulations. Mixture designs can be used to describe the response (tablet properties) as a function of the... 

Further examples of alkylation of imidazole derivatives were recently reported by Galous et a/.145 The nature and importance of significant factors in phase transfer alkylation of pyrazole was studied by Elguero et al.146 Their conclusions are equivalent to those of Dehmlow and Lissel.147 The optimization method used by Elguero et al. (several parameters at a time) is different from the conventional procedure (one parameter at a time) and will probably find applications in the future for optimization of organic syntheses. 

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