# Response Surfaces

One of the most effective ways to think about an optimization is to visualize how a system’s response changes when we increase or decrease the levels of one or more of its factors. We call a plot of the system’s response as a function of factor levels a response surface.

The simplest response surface has a single factor, which we represent graphically in two dimensions by placing the response on the y-axis and the factor’s levels on the x-axis. The calibration curve below is an example of a one-factor response surface, which is represented by the equation

A = 0.008 + 0.00896CA

where A is the absorbance and CA is the analyte’s concentration in ppm.

For a two-factor system, the response surface is a plane in three dimensions in which we place the response on the z-axis and the factor levels on the x-axis and the y-axis. In the illustration below, (a) shows a pseudo-three dimensional wireframe plot for a system obeying the equation

R = 3.0 – 0.3A + 0.020AB

where R is the response, and A and B are the factors. We can also represent a two-factor response surface using the two-dimensional level plot in (b), which uses a color gradient to show the response on a two-dimensional grid, or using the two-dimensional contour plot in (c), which uses contour lines to display the response surface.