Fixed Sized Simplex

The most common, and easiest to understand, method of simplex optimisation is called the fixed sized simplex. It is best described as a series of rules. 

The main steps are as follows, exemplified by a two factor experiment. 

Define how many factors are of interest, which we will call k. 

Perform k + 1(=3) experiments on the vertices of a simplex (or triangle for two factors) in factor space. The conditions for these experiments depend on the step size. This defines the final resolution of the optimum. The smaller the step size, the better the optimum can be defined, but the more the experiments are necessary. A typical initial simplex using the step size above might consist of the three experiments, for example 

Rank the response (e.g. the yield of rate of the reaction) from 1 (worst) to k + 1 (best) over each of the initial conditions. Note that the response does not need to be quantitative, it could be qualitative, e.g. which food tastes best. In vector form the conditions for the nth response are given by xn, where the higher the value of 

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Because the size of the simplex remains constant during the search, this algorithm is called a fixed-sized simplex optimization. Example 14.1 illustrates the application of these rules. 

Find the optimum response for the response surface in Figure 14.7 using the fixed-sized simplex searching algorithm. Use (0, 0) for the initial factor levels, and set the step size for each factor to 1.0. 

Progress of Fixed-Sized Simplex Optimization for Response Surface in Figure 14.10... 

Progress of a fixed-sized simplex optimization for the response surface of Example 14.1. The optimum response at (3, 7) corresponds to vertex 25. 

This experiment describes a fixed-size simplex optimization of a system involving four factors. The goal of the optimization is to maximize the absorbance of As by hydride generation atomic absorption spectroscopy using the concentration of HCl, the N2 flow rate, the mass of NaBH4, and reaction time as factors. 

The progress of a fixed sized simplex is illustrated in Figure 2.38. 

For the modified simplex, step 4 of the fixed sized simplex (Section 2.6.1) is changed as follows. A new response at point xlesl is determined, where the conditions are obtained as for fixed sized simplex. The four cases below are illustrated in Figure 2.39. 

Xnew = Xtest = C C X1 as in die normal (fixed-sized) simplex. 

Figure 4.17 Fixed-size simplex according to Nelder and Mead along an unknown response surface.

Hgure 5 Progress of a typical fixed size simplex. 

The progress of a fixed sized simplex is illustrated in Figure 5. Many elaborations have been developed over the years. One of the most important is the k + rule. If a vertex has remained part of the simplex for k + 1 steps, perform the experiment again. The reason for this is that response surfaces may be noisy, so... 

Then perform another experiment at x ew and keep this new experiment plus the k (=2) best experiments from the previous simplex to give a new simplex. Rule 5 of the fixed sized simplex still applies, if the value of the response at the new vertex is less than that of the remaining k responses, return to the original simplex and reject the second best response, repeating the calculation as above. 

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