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A Desirability Response is a response calculated from one or more responses (a multivariate analysis) which are themselves measured in the natural units of the process. Each response is scaled to transform the original response to a range of zero to one for a specified range of the original response values. The scaled values are given a measure of importance by being raised to a power.
A Desirability Response may be a simple linear transform using one or more responses to reflect a single parameter of interest, such as cost in dollars. More generally it may use a nonlinear combination of responses to form a single response. The application of upper and lower limits defines a nonlinear or step valuation of the response. Raising the scaled values to a power makes the response more or less valuable as it deviates from a Target.
The term Desirability Response can also be specifically applied to a nonlinear algebraic weighting function. Each data response may be individually transformed with unique limits and importance powers.
where: D(I, J) is the transformed desirability for the Ith row of the Jth response, which is computed from Lower Limit (LL), Target (Targ) and Upper Limit (UL) values in the units of the Jth response. The exponents S and T are measures of importance which typically range from 0.1 to 10.0 .
The individual transformed responses are combined by calculating their geometric mean.
The composite Desirability Response is calculated as,
D = (d1 * d2 * d3 * … * dm)1/m
The effect of the limits is to make the factor desirability zero outside the limits. The target value has the maximum desirability of 1.0 . The effect of S and T are to weight the importance of the factor within its desired range. S or T equal to 1 is for linear weighting.
Desirability Responses may then be plotted, used in a regression, etc..
The plot of the Desirability Function shown below is for a factor with a Target value of 0 with scaled response values that lie between a lower limit of -1 and 0. A corresponding plot would describe the weighting for the region between the Target and an upper limit. The importance powers range from 0.01 to 100.0
High values of the exponent tend to value highly those response values near the Target; low values of the exponent cause a wider band around the Target to be desired almost as well as the Target. For example, a scaled response of -0.975, almost at the lower limit, with an exponent of 0.01 has a desirability of 0.96.
If any value below the target is about as desirable as the Target, an artificial LL may be set very far below the Target, with a very low value of the importance exponent. Similarly, if a response below the Target is strongly disliked the LL should be set close to the Target with a high value for the exponent. The LL may be set at the same value as the Target in which case the desirability of all response values below the Target will be zero.
Corresponding arguments apply to the upper region.