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The Simplex Optimization Methods

The Modified Simplex Method

The modified simplex method has much in common with the basic method, but can adjust its shape and size depending of the response in each step. This method is also called the variable-size simplex method. Several new rules are added to the basic simplex rules. These new rules make the simplex: 

• Expand in a direction of more favorable conditions.

Or

• Contract if a move was taken in a direction of less favorable conditions.

The procedures for expansion and contraction enable the modified simplex both to accelerate along a successful track of improvement and to home in on the optimum conditions. Therefore the modified simplex will usually reach the optimum region quicker than with the basic method and pinpoint the optimum levels more closely. Moves in a typical optimization sequence are easy to show for two control variables (the response is shown separately).

An example of a typical optimization sequence with the modified simplex method. Change in the levels for two control variables.

An example of a typical optimization sequence with the modified simplex method. Change in the response.

The degree of contraction depends on how unfavorable the new response is. The next figure illustrates the different moves with the modified simplex method.

Different simplex moves from the rejected trial condition (W). R = reflection, E = expansion, C+ = positive contraction and C- = negative contraction.

The optimization algorithm used in the MultiSimplex software contains some generally accepted modifications to the original version of the modified simplex method. The calculations in the MultiSimplex modified 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 N_w for the second least favorable trial (i.e. next-to-the worst).

 

The different projections away from the rejected trial are calculated according to the following formulas:

Where

• W is the rejected trial.
• is the centroid of the remaining face/hyperface, i.e. the average levels for the remaining trials.
• a is the reflection coefficient.
• g is the expansion coefficient.
• b⁺ is the positive contraction coefficient.
• b^- is the negative contraction coefficient.

 

Literature

Nelder, J. A., Mead, R. A simplex method for function minimization. Computer Journal 7(1965) 308-313.

Åberg, E. R., Gustavsson, A. G. T. Design and evaluation of modified simplex methods. Analytica Chimica Acta 144(1982) 39-53.

Betteridge, D., Wade, A. P., Howard, A. G. Reflections on the modified simplex - II. Talanta 32(1985):8B 723-734.

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.

1. Simplex Introduction
2. The Basic Simplex Method
3. Evolutionary Operation
4. Mixture Optimization

Back to Optimization Methods Introduction