What is a must for a desirability function d?

Step 1+2: Target value and maximisation problems can be formulated.
Parameters useful for target value problems are:

d monotonically increasing in $(- \infty, T)$ and decreasing in $(T, \infty)$ .

Guarantees pareto-optimality of any desirability-optimum!

Nice to have: weights $\beta_l, \beta_r$ for deviations to the left or the right of T.

Step 3: Use some kind of mean value.

Most popular choice is the geometric mean.

Important feature: D(X) = 0, if any d_i(X) = 0

"If one of the properties is completely unacceptable, the product as a whole is unacceptable."

$DI(P) := \sqrt[Z]{\prod_1^Z d_i(Y_i)} = \sqrt[Z]{\prod_1^Z d_i(f_i(X_1,X_2, \ldots, X_F,\epsilon_i))}$

Z-th root is needed for comparability of single and overall results.

Alternative: min D

$\begin{displaymath}D(P) := \max_{{\bf X}} \min_{i = 1, \ldots, Z} d_i({\bf X}). \end{displaymath}$

"A product is only as good as its worst property."

No alternative, but often used in practice: arithmetical mean.
This allows partly unusable products.

Dipl.-Stat. Detlef Steuer
1999-10-04