On the use of desirability functions

The problem of many responses of approximately equal interest is often met in industrial synthesis. Cost, quality specifications, overall yield as well as pollution due to waste are necessary to control, and it is sufficient that one of the responses fails to meet the requirements for an overall result to be poor. In such cases it is possible to use a technique by which all responses are weighed together into one criterion which is then optimized. First, it is necessary to clearly define, what the desired result is for each response. The measured value of the response is then scaled to a dimensionless measure of the desirability, dj, of the response jj. The scaling is done so that dj is in 

It is seen, that a high value of D is obtained only if all individual desirabilities d are high. If one d is poor, this is enough to give an overall poor desirability. If all d are equal, then D = d. The geometric mean is therefore seen to meet the requirements of an overall desirability function. The problem is then to define the individual desirabih ties. 

It may be required that a response y should be in a specified interval, 

Such scaling of d as shown in Fig. 12.24 would give two possible overall results success or failure. At best, this can be used for classification of the experimental conditions as being useful or useless. Thus, a discrete scaling like this cannot be used for optimization. 

A smooth and continuous function to define d is necessary, and such functions can be obtained by mathematical transformation of the measured response. One way to do this has been suggested by Harrington [20], and this procedure is given here. Other ways of scaling have been given by Derringer and Suich.[21] See also a discussion in the book by Box and Draper.[22] 

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