Sequential Simplex Optimization

One of the most popular methods of EVOP is sequential simplex optimization. The simplex is a geometric figure with the number of vertexes equal to the number of factors plus one. A simplex with one factor is a line, a simplex with two factors is a triangle, etc. The simplex is a simplistic model of the surface. a new simplex is formed by eliminating the vertex with the worst response and replacing it by projecting it through the average coordinates of the remaining vertexes. A new experiment is performed with the factor levels determined by the coordinates of the new vertex, and the process is repeated until an optimum response is found. Using the sequential simplex method optimization has two advantages over using factorial designed experiments for EVOP.

1. The number of trials in the initial simplex is k+1 (k is the number of experimental factors), while a factorial approach has at least 2^k and possibly 3^k or 4^k.
2. Only one new trial is required to move to a new area in the space defined by the factors; while a factorial design requires at least 2^k-1 trials. An example of a sequential simplex optimization is shown in the figure below.

To compute the coordinates of the new vertex (R), refer to the following table, where B is the vertex with the best response, N is the vertex with the next best response, and W is the vertex with the worst response. This table shows an example for a 3 factor optimization (k = 3). The logic is the same for any number of factors.

The optimum point is never reached in the figure above because the size of the individual simplex is too large. If the simplex size is too small, an excessive number of steps is required to reach the optimum point. This problem has been solved by using a variable size simplex. Instead of a simple reflection (R), there are other options:

• double the length of the reflection (E),
• reduce the length of the expansion by 50% (Cr),
• produce the reflection in the opposite direction (W), and
• produce the reflection in the opposite direction but at 50% of the normal length (Cw).

The additional rules for determining the location of the new vertex are:

• If the response for the new vertex (R) is better than the response for the next best vertex (N), but worse than the response of the best (B) vertex, then use R as the new vertex.
• If the response for the new vertex (R) is better than the response of the best vertex (B), then compute E = P + 2(P-W).
• If the response of E is better than the response of B, use E as the new vertex; if not, use R as the new vertex.
• If the response for the new vertex (R) is worse than the response of the next best vertex (N) but better than the response of the worst vertex (W), then compute Cr = P + 0.5(P-W) as the new vertex.
• If the response for the new vertex (R) is worse than the response of the worst vertex (W), then compute Cw = P - 0.5(P-W) as the new vertex.

Example

Consider the parameter Y determined from the function Starting with the vertices

Y =  40A + 35B - 15A² - 15B² + 25AB

Solution

Starting with the vertices

The response Y is maximized using a variable-step sequential simplex as shown below.

After 16 step the best response is 279. The optimum value is 281 when A = 7.54 and 7.46.