Automatic recommendation of a design

In this section you can find a detailed description of the method that we use to automatically select the best design from a small set of competing designs.

Utopia method for selecting automatically a design

The starting point for our algorithm is a set of designs $D$ which are characterized by a number $h$ of numerical attributes of interest, which are specified in the set $A$.

We define the utopia point u as the vector of attribute values of an ideal design which is given by $u := (a_1, . . . , a_h)$, where $a_i$ is the best value for the $i$th attribute in $A$ over all designs in the set $D$. Then, we select the design in $D$ which is the closest to u using the usual euclidean distance, after normalizing all attributes in $A$. For further information on the utopia method, we refer to Marler and Arora (2004) ¹

This process outputs one design, unless several designs are at the same distance from $u$. Note that the utopia method implicitly gives the same weight to all attributes in $A$ and works for any cardinality of $D$.

Example

We illustrate how the method works by means of an example.

Consider a set of four designs which are characterized by 5 numerical statistical quality parameters. The designs and the numerical attributes are displayed in Table 1.

For some of the attributes a higher value is desired (powers and G-efficiency). For others, a lower value is preferred (number of tests and 4th order correlation). Given this information, which we call directions of improvements, the following transformation is applied to some of the columns in Table 1. The columns that are transformed are those corresponding to statistical quality parameters for which lower values are preferred, this is, the second and third columns which correspond respectively to the number of experimental tests or runs and the 4th order correlation. The transformation consists in multiplying all values in those columns by $-1$. By performing this transformation, all attributes become the bigger the better. The maximum value for each column is highlighted. Table 2 shows the data of Table 1 after applying this transformation.

We then proceed to normalize the data on each column so that the minimum value becomes 0 and the maximum becomes 1. The results after normalization are displayed in Table 3. The maximum value for each column is highlighted.

Next, we obtain the utopia point. It is clear that, after normalization, the utopia point has all its coordinates equal to 1.

The final calculation consists of obtaining the Euclidean distance of each design to the utopia point, which is displayed in Table 4.

For this example, Design 4 is the recommended one, as it is the closest to the utopia point.

We can apply the utopia selection also to a subset of the numerical attributes. Say that we mostly care about the number of tests, the 4th order correlation and the power to detect quadratic effects. Then, we can calculate the distances to the utopia point for these three attributes, obtaining the results in Table 5. In this case, Design 1 would be the recommended design, which makes sense as it is the best design in what respects the number of tests and the 4th order correlation.