Derringer desirability function

RSM Derringer desirability function. Robustness testing using a multivariate approach... 

The problem of working with several criteria is that in general they are conflicting, i.e., the best for one criterion is not the best for another. In these cases one needs to somehow balance the criteria in order to have the best possible for each of the criteria at the same time. Several methods have been proposed to do this and, among others, one can use utility functions, the Pareto, Electre, and Promethee methods, or the Derringer desirability function. Since the Pareto concept and the Derringer desirability function have elicited particular interest in chemistry, this section will be restricted to them. 

Figure 17 Possible two-sided transformations of the response values for the Derringer desirability function approach...

Jimidar, M., Bourguignon, B., and Massart, D. L. (1996). Application of Derringer s desirability function for the selection of optimum separation conditions in capillary zone electrophoresis. J. Chromatogr. A 740(1), 109-117. 

MCDM methods are applied when at least two responses need to be optimized simultaneously. Different approaches can be distinguished, for example, window programming, threshold approaches, utility functions. Derringer s desirability functions, Pareto optimality methods, Electre outranking relationships, and Promethee (7). In this chapter, only the Pareto optimality methods (7, 117, 118) and Derringer s desirability functions (7, 119, 120) will be discussed. 

A second MCDM approach is the use of Derringer s desirability functions. In this approach, all responses are transformed on the same scale and combined to one response, D, which then should be maximized. Each response is transformed on a scale between 0, representing the most undesirable outcome, and 1, representing the most desirable situation. The values of the transformed responses are called desirabilities. Different transformations are used, depend-... 

FIGURE 2.20. Derringer s desirability functions the response is optimal when (a) maximized, (b) minimized, and (c) at a given value. 

Derringer and Suich (10) described a way of overcoming this difficulty. In the optimization, each response i is associated with its own partial desirability function (d,). This varies from 0 to 1, according to the closeness of the response to its target value. For the friability, we would like as low a value as possible. The target is therefore 0% friability and the desirability at this point is equal to 1. If no formulation with a friability of more than 2% can be considered acceptable, than the desirability is equal to zero for all values of 2% and over. Between 0% and 2% the desirability decreases linearly or in a convex or concave form. 

The Derringer and Suich method is based on the definition of a desirability function for each response, with values restricted to the [0,1] interval. Zero stands for an unacceptable value, while one is assigned to the most desirable value. The nature of the function depends on the objectives of the experiment, as we shall see. 

There are many ways to produce suitable desirability functions, one of which is explained in (Derringer Suich, 1980). Any function that gives the 1 value for a perfect response and the value 0 for an unacceptable product and continuously values between 0 and 1 for responses whose goodness is in-between unacceptable and perfect can be used. One of the simplest... 

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