Desirability Details
Myers and Montgomery (Response Surface Methodology, p. 244) describe a multiple response method called desirability. The method makes use of an objective function, \(D(X)\), called the desirability function. It reflects the desirable ranges for each response (\(d_i\)). The desirable ranges are from zero to one (least to most desirable, respectively). The simultaneous objective function is a geometric mean of all transformed responses:
\(D = (d_1 \cdot d_2 \cdot...\cdot d_n)^{\frac{1}{n}}=\left(\displaystyle\prod_{i=1}^{n}d_i\right)^{\frac{1}{n}}\)
Where \(n\) is the number of responses in the measure. If any of the responses or factors fall outside their desirability range, the overall function becomes zero.
For simultaneous optimization each response must have a low and high value assigned to each goal. On the worksheet, the “Goal” field for responses must be one of five choices: “none”, “maximum”, “minimum”, “target”, or “in range”. Factors will always be included in the optimization, at their design range by default, or as a maximum, minimum of target goal. The meanings of the goal parameters are:
Maximum:
\(d_i = 0\) if response < low value
\(0 < d_i < 1\) as response varies from low to high
\(d_i = 1\) if response > high value
Minimum:
\(d_i = 1\) if response < low value
\(1 > d_i > 0\) as response varies from low to high
\(d_i = 0\) if response > high value
Target:
\(d_i = 0\) if response < low value
\(0 < d_i < 1\) as response varies from low to target
\(1 > d_i > 0\) as response varies from target to high
\(d_i = 0\) if response > high value
Range:
\(d_i = 0\) if response < low value
\(d_i = 1\) as response varies from low to high
\(d_i = 0\) if response > high value
The \(d_i\) for “in range” are included in the product of the desirability function \(D\), but are not counted in determining \(n\) in the formula \(D =(\prod{d_i})^{\frac{1}{n}}\).
If the goal is none, the response will not be used for the optimization.
Goals can be set for any number responses, factors and/or components provided at least one response is included in the goal set.