The basic simplex method is easy to understand and apply. The optimization begins with
the initial trials. The trial conditions are spread out efficiently. The number of initial
trials is equal to the number of control variables plus one. These initial trials form the
first simplex. The shapes of the simplex in a one, a two and a three variable search
space, are a line, a triangle or a tetrahedron. A geometric interpretation is difficult
with more variables, but the basic mathematical approach outlined below can handle the
search for optimum conditions.
Geometric interpretation of lower dimensional simplices.
The basic simplex algorithm consists of a few rules:
The first rule is to reject the trial with the least favorable response value in the
current simplex.
A new set of control variable levels is calculated, by reflection into the control
variable space opposite the undesirable result. This new trial replaces the least
favorable trial in the simplex. This leads to a new least favorable response in the
simplex that, in turn, leads to another new trial, and so on. At each step you move away
from the least favorable conditions. By that the simplex will move steadily towards more
favorable conditions.
The second rule is never to return to control variable levels that have just been
rejected.
The calculated reflection in the control variables can also produce a least favorable
result. Without this second rule the simplex would just oscillate between the two control
variable levels. This problem is nicely avoided by choosing the second least favorable
condition and moving away from it.
An example of a typical optimization sequence with the basic simplex
method. Change in the levels for two control variables with the response marked as
contours.
Besides the two main rules, two more rules are also used.
Trials retained in the simplex for a specified number of steps are reevaluated. The
reevaluation rule avoids the simplex to be stuck around a false favorable response.
Calculated trials outside the effective boundaries of the control variables are not
made. Instead a very unfavorable response is applied, forcing the simplex to move away
from the boundary.
The calculations in the MultiSimplex basic simplex algorithm are outlined in the flow
chart. For each simplex the following labels are used: W for the least
favorable trial or the trial being rejected, B for the most favorable trial
and Nw for the second least favorable trial (i.e. next-to-the
worst).
Spendley, W., Hext, G. R., Himsworth, F. R. Sequential application of simplex
designs in optimisation and evolutionary operation. Technometrics 4(1962):4 441-461.
Suggested further reading:
Sequential Simplex Optimization. A Technique for Improving Quality and Productivity
in Research, Development, and Manufacturing by Walters, Parker, Morgan and Deming, CRC
Press 1991.
