A simple algorithm for optimizing the response for a system is to adjust independently each factor. Consider a response that depends on two factors. We begin by optimizing the response for one factor by increasing or decreasing its value, holding constant the value of the second factor. We then vary the value for the second factor, holding the value of the first factor at its previously determined optimum value. We can stop this process, which we call a one-factor-at-a-time optimization, after a single cycle or run additional cycles until we reach the optimum response or until the response exceeds an acceptable threshold value.
A one-factor-at-a-time optimization is an effective, although not necessarily an efficient experimental design when the factors are independent. Two factors are independent when changing the level of one factor does not influence the effect of changing the other factor’s level, as illustrated here where the parallel lines show that the level of factor B does not influence factor A’s effect on the response.
Mathematically, two factors are independent if they do not appear in the same term in the equation describing the response surface. The response surface below, for example, is for the equation
R = 2.0 + 1.2A + 0.48B – 0.03A2 – 0.03B2
For independent factors, a one-factor-at-a-time optimization quickly and efficiently finds the global optimum, as illustrated by the orange lines in (b).
Unfortunately, factors usually do not behave independently. Consider, for example, the figure below, which shows a dependent relationship between the a factors.
Dependent factors are said to interact and the response surface’s equation includes an interaction term containing both factors A and B. For example, the final term in the following equation accounts for the interaction between factors A and B.
R = 5.5 + 1.5A + 0.6B – 0.15A2 – 0.0245B2 – 0.0857AB
and yield the response surfaces shown here.
The progress of a one-factor-at-a-time optimization for dependent factors is illustrated by the orange line in (b) above. Although the optimization is effective, in that it finds the global optimum, it is less efficient than that for independent factors. In this case it takes four cycles to reach the optimum response of (3, 7) if we begin at (0, 0).