RSM Simplified by Mark J. Anderson & Patrick J. Whitcomb

Author:Mark J. Anderson & Patrick J. Whitcomb
Language: eng
Format: epub
Publisher: CRC Press

Figure 6A.1 Simplex with the evaluation of response.

The labels describe hypothetical rankings of overall desirability in the 2D factor space. Here are the rules according to the originators of simplex optimization (Spendley et al., 1962):

1. Construct the first simplex about a random starting point. The size can vary, but a reasonable side length is 10% of the coded factor range.

2. Compute the desirability at each vertex.

3. Initially, reflect away from the worst vertex (W). Then, on all subsequent steps, move away from the next-to-worst vertex (N), from the previous simplex.

4. Continue until the change in desirability becomes very small. It requires only function evaluations, not derivatives.

Figure 6A.2 may help a bit by picturing a move made from an initial simplex for three factors—called a “tetrahedron” in geometric parlance.

The use of next to worst (N) from the previous simplex ensures that you won’t get stuck in a flip-flop (i.e., imagine going down rapidly in a rubber raft and getting stuck in a trough). The last N becomes the new W, which might better be termed as a “wastebasket” because it may not be worst. If you’re a bit lost by all this, don’t worry—as you can see in Figure 6A.3, it works!

This hypothetical example with two factors, temperature and catalyst, shows how in a fairly small number of moves, the simplex optimization converges on the optimum point. It will continue going around and around the peak if you don’t stop it (never fear, the programmers won’t allow an infinite loop like this to happen!).

So far, we’ve shown simplexes of a fixed size, but it’s better to allow them to expand in favorable directions and, conversely, contract when things get worse instead of better. Another few lines of logic, which we won’t get into, accomplish this mission of making variable-sized simplexes (Nelder and Mead, 1965). Figure 6A.4 gives you a view of how this works for optimizing a response.

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