After using the Steepest Ascent method to find the optimum location in terms of our factors, we can now go directly to the second order response surface design. A favorite design that we consider for a second order model is referred to as a central composite design.
Here is an example in two dimensions: Example 11.2 in the text. We have \(2^k\) corner points and we have some number of center points which generally would be somewhere between 4 and 7, (five here). In two dimensions there are 4 star points, but in general there are \(2^k\) star points in \(k\) dimensions. The value of these points is something greater than 1. Why is it something greater than 1? If you think about the region of experimentation, we have up to now always defined a box, but if you think of a circle the star points are somewhere on the circumference of that circle, or in three dimensions on the ball enclosing the box. All of these are design points around the region where you expect the optimum outcome to be located. Typically the only replication, in order to get some measure of pure error, is done at the center of the design.