Simplex initial experiments

Purpose To determine, from eight initial experiments performed under certain conditions, whether the three controlled parameters have an effect on the measurement, and which model is to be used. This factorial approach to optimization is an alternative to the use of multidimensional simplex algorithms it has the advantage of remaining transparent to the user. 

In the simplex method, die number of initial experiments conducted is one more than the number of parameters (temperature, gradient rate, etc.) to be simultaneously optimized. The conditions of the initial experiments constitute the vertices of a geometric figure (simplex), which will subsequently move through the parameter space in search of the optimum. Once the initial simplex is established, the vertex with the lowest value is rejected, and is replaced by a new vertex found by reflecting the simplex in the direction away from the rejected vertex. The vertices of the new simplex are then evaluated as before, and in this way the simplex proceeds toward the optimum set of conditions. 

Section 5.3 describes sequential methods of optimization, in particular the Simplex method. In sequential methods the optimization procedure starts with some initial experiments, inspects the data and defines the location of a new data point which is expected to yield an improved chromatogram. The idea is to approach the optimum step by step in this way. 

In contrast to the simultaneous optimization procedures described in the previous section, the Simplex method is a sequential one. A minimum number of initial experiments is performed, and based on the outcome of these a decision is made on the location of a subsequent data point. This simplest form of a sequential optimization scheme can be characterized by the path 1012 in figure 5.4. 

The number of initial data points is one more than the number of parameters considered in the optimization process. These initial experiments define a geometrical figure in the parameter space which is called a Simplex. A two-dimensional Simplex is a triangle (often equilateral). A three-dimensional Simplex is a tetrahedron. The description of Simplexes in more dimensions is somewhat more difficult to envisage, but is mathematically straightforward. 

Restarting the Simplex from different initial experiments will decimate problems 2. and 3. above, but will aggravate point 1. Because of the likelihood that the Simplex optimization will lead to a local optimum, the use of an initial coarse Simplex in order to find a suitable area in which an experimental design can be located [512] cannot be recommended. 

Initial experiments (a, b and c) on the edge of a simplex two factors, and the new conditions if experiment a results in the worst response... 

Sequential optimisation methods are used for multi-parameter optimisation. The simplex method starts with some initial experiments, evaluates from them the values of a sum optimisation criterion (COF), on the basis of these results determines the next combination of operation parameters to be used for running a new chromatographic experiment and compares the value of the COF obtained from the new experiment with the old one. On the basis of this prediction, a new combination of the operation parameters is calculated which is expected to yield an improved value of the COF, the separation is run at these new conditions and the procedure is repeated until maximum COF with no further improvement is eventually obtained, for which — hopefully — the optimum combination of operation parameters has been obtained (Fig. 1.22). Any combination of operation parameters can be optimised in this way and no knowledge about the nature of the chromatographic process is necessary ( black-box philosophy). Some HPLC control systems allow the simplex optimisation to run unattended. 

The method considers only two factors, only three initial experiments are required to perform the simplex process and obtain the maximum A/ /. ,i , which can be selected from the nine preliminary experiments. 

Figure 3.6. A Simplex optimization of a two-factor system. Numbers give the order of experiments. The first three experiments (points 1, 2, and 3) define the initial Simplex, and dashed-line arrows indicate which points are dropped in favor of new factor values.

It is clearly beyond the scope of this chapter to consider further the selection of which variables to use in the simplex optimization. To summarize our own relatively limited experience, however (boxes in Table IV represent combinations examined to date), we recommend the following For a relatively simple separation, begin with a two-parameter simplex that includes either initial pressure (or density), using as many characteristics of the analytes and/or sample matrix to logically deduce which remaining variable to optimize. For a more complex separation, or one in which little is known about the sample, try a 4 or 5-variable simplex that includes the initial pressure and pressure gradient (or initial density and density gradient) as optimization variables. 

Besides these three problems, one should also know how to switch from simplex to a second-order design that may describe the optimum area. This is the subject of sect. 2.5.4. The first problem in simplex optimization consists of constructing the matrix of a design of experiments for initial simplex where coordinates of experimental points-vertices are given. In solving this problem, different orientations of initial simplex to the coordinate system are possible. A simplex center is mostly set in the coordinate beginning, while the distance between simplex vertices (simplex sides) has a coded value of one. Simplex is, as a rule, oriented in a factor space in such a way that vertex l>k+I lies on the xk axis, while other vertices are distributed symmetrically with respect to coordinate axes. Simplexes of such a construction are shown in Figs. 2.50 and 2.51. 

Assume we have to do optimization of a phenomenon that is defined by four factors (k=4). We use Table 2.209 to apply simplex optimization and define the initial simplex. To determine the operational matrix, we should know factor values in the center of experiment and their variation intervals. These data are to be found in Table 2.210. Real factor values are obtained from relation (2.59) ... 

If the most inconvenient response value appears in the new vertex which is then rejected, the simplex is returned to the initial point, and we call this simplex swaying, Fig. 2.53C. We may avoid this simplex swaying by returning to the initial vertex and rejecting the vertex that is the second in a sequence of inconvenient response values. If even then we do not reach a satisfactory result, we again return to the initial vertex and reject the third one. In practice, we may be faced with false simplex movement to optimum as a consequence of large error of experiment. In most cases, however, we should not pay great attention to an error of experiment since it... 

A fourth-order D-optimal design is produced with reference to pseudocomponents Zi Z2 and Z3 - Table 3.42. The pseudocomponents satisfy the principal condition for Scheffe s designs. The conversion to initial components at any point within the local simplex studied is carried out from Eq. (3.84). According to this design, an experiment is run with mixtures, each observation being repeated twice. Using Eqs. (3.109)-(3.113) the coefficients of fourth-order regression equation are calculated in pseudocomponents... 

It is seen in figure 5.7 that point M is not exactly located at the optimum (point O). A decrease in the size of the initial Simplex will imply that the optimum will be approached more closely, but also that the number of experiments will increase further. Since the simple optimization described in figure 5.7 has already required 17 experiments, the latter prospect is not very attractive. 

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... 

When the experiments required by the initial simplex regular plan are completed then we eliminate the point that produces the most illogical or fool response values by building the image of this point according to the opposite face of the simplex, we obtain the position of the new experimental point. 

In this table, we can observe that the 7th experiment has been produced and its corresponding y value has been given. We can notice that, in simplex 123567, point number 3 is the less favourable point for the process (in this case it is the point with the lowest yield). It should therefore be eliminated. Now we can proceed with the introduction of the temperature as a new process factor. In the previous experiments, the temperature was fixed at z° = 45 °C. Initially, we consider that z = 45 °C and we select the variation interval to be Azg = 15 °C. In this situation, if we apply Eq. (5.95), we have x = obviously x = 0. In order... 

Let us consider the case of two variables, x and xi. An algorithm describes the initial simplex to be performed (Fig. 43.2). By performing experiments 1-3, described by the initial simplex, and recording their responses, the next set of experiments can be described. If we obtain the lowest response for experiment 2, it can therefore be assumed that a higher response would be obtained in the opposite direction. By reflecting point 2, we can obtain point 4. By performing the experiment described by point 4 we obtain its response, thereby perpetuating the simplex. 

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