Partial desirability

Partial responses transformed into the non dimensional scale are marked du(u=1.2,...,n) and called partial desirability or individual desirability. As shown in Table 2.6 the desirability scale has the range from 0.0 to 1.0. Two characteristic limit values for quality are within this range 0.37 and 0.63. The 0.37 value is approximately l/e=0.36788, where e is the basis of the natural logarithm, and 0.63 is 1-1/e. 

Assume we have an experiment where we dispose with specifications with one or two limit values for each partial response. For those values outside the limit values we have du=0, and within them d -1. If yrnin is the lower limit value of the specification and ifyu>ymin then the partial desirability Junction for a one-sided limitation is ... 

Figure 2.5 Partial desirability function with one- and double-sided limitation...

Here we have a very simple classification on acceptable and unacceptable quality, which is rarely met in practice. Transformation of partial responses into a partial desirability, in a large number of cases uses Table 2.6 and desirability (2.13). 

For one-sided limitations yL1ymin, partial desirability, limited on one side, is shown in Fig. 2.6. 

After choosing the desirability scale and after the transformation of partial responses into partial desirability it is possible to approach constructing the general response D, which is called Harrington s over all desirability or Harrington s general response. To generalize or switch from du to D is possible by the formula ... 

Transformed partial responses into partial desirability are shown in Table 2.7. Following Harrington s general response ... 

Table 2.7 Partial responses, partial desirability and overall desirability...

Among the response requirements of a research subject that have to be met in the first place is that it has to be quantitative. A researcher usually keeps to this requirement, however there are situations when it cannot be met, and the researcher has to deal with qualitative responses. Due to the fact that in the case of qualitative responses the efficiency of experimental research is reduced, one should try to transform these responses into quantitative ones. For this, one may use the transformation of qualitative response by desirability scale into partial desirability. 

The problem of multiple responses Partial desirability functions Examples of multi-criteria optimization... 

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. 

We will examine the different kinds of partial desirability functions available using the same example of a granulation process study (8) that has already been used to demonstrate graphical analysis and optimization. 

It is assumed that the experimenter, having carried out his experiments and analysed the results, will now have a satisfactory mathematical model for each response that is to enter into the global optimization. He can therefore calculate a partial desirability for each response in all parts of the domain, and obtain from these an overall desirability D. The problem becomes one of numerical optimization of the function D within the domain. 

In this case the cohesion index required maximization. Values below 600 were considered unacceptable. The target value was 1000 and it was not considered that increasing the index above this value would improve the result in any way. The resulting partial desirability function d, for the response is shown in figure 6.9a. 

The friability of the tablets was to be minimized. Values above 0.5% (the threshold) were to be rejected, but all friabilities below 0.1% (the target value), on the other hand, were considered completely acceptable. The partial desirability function dj for the response yj is shown in figure 6.9b. 

A value s = 3 gave the curve shown in figure 6.10a. A similar relationship was selected for the tablet hardness (figure 6.10b), where the minimum value 11 kP was penalized still further, as a higher exponent s = 5 was selected. In the case of the linear partial desirability function, the exponent s is unity. 

In combining partial desirability functions, it is possible to take a weighted mean. However it is far more frequent to use a geometric mean of the desirabilities, of the form ... 

This allows optimization to take into account the relative importance of each response, while selecting the most appropriate form of the partial desirability function. Equation 6.5 is the special case of equation 6.6 for equal weightings and is the form most often used at present. It has the disadvantage that weighting can only be done by adjusting x. 

Response Partial desirability y (min) (target) yi y (max) 1 Exponent s i lower upper... 

We set up partial desirability functions, as in table 6.7, using the same limits as the authors (7) for the size and slightly wider limits for the yield. Allowance also had to be made for the fact that the response data were transformed. The range for the size was relatively narrow, so the transformation would have little effect on the partial desirability function. On the other hand, the wide range for the yield means that smaller yields would tend to be penalised less than if there was no transformation. For this reason we increased the value of the exponent to s = 5 (c.f. figure 6.10b), in order to favour higher yields and compensate for this effect. 

Response Partial desirability Goal y (max) 1 Predicted response Partial desirability... 

If the responses are transformed, remember that the desirability functions are on the transformed responses. It may be necessary to compensate by adding an exponent to the partial desirability function. 

The desirability method described in section in of chapter 6 may also be used. The minimum hardness allowed is 80 N. The tablets are considered better as the hardness increases to 150N. Similarly, the disintegration time is minimized. The maximum allowable disintegration time is 5 minutes. For times of 1 minute and below, the partial desirability function is equal to 1. 

Possibility of weighting each partial desirability ftmction within the global desirability... 

Flavoring matter, especially bitter substances, which are partially desired (coffee) but can also cause an off-taste, e. g., in grilled meat or fish (roasting bitter substances). 

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