The basic simplex

There exist several simplex methods. In this chapter, we will discuss three of them, in increasing order of complexity the basic simplex, the modified simplex and the super-modified simplex. The more sophisticated methods are able to adapt themselves better to the response surface studied. However, their construction requires a larger number of experiments. In spite of this, the modified and super-modified simplexes normally are able to come closer to the maximum (or minimum if this were of interest) with a total number of experiments that is smaller than would be necessary for the basic simplex. In this chapter, we will see examples with only two or three variables, so that we can graphically visualize the simplex evolution for instructive purposes. However, the efficiency of the simplex, in comparison with univariate optimization methods, increases with the number of factors. 

the basic simplex is the simplest of all. This simplex is always a regular geometrical figure, whose form and size do not vary during the optimization process. For this reason it may not be very efficient (Spendley et al., 1962 Deming, 1981). With two factors, the simplex is 

The basic idea of the methods discussed in this chapter is to displace the simplex over the response surface in such a way as to avoid regions with imsatisfactory responses. In the present example, since we obviously would like to obtain a maximum yield, we should stay away from points that correspond to low jdelds. This is done using five rules. 

Rule 1 - The first simplex is determined performing a number of experiments equal to the number of factors plus one. The size, position and orientation of the initial simplex are chosen by the researcher, taking into accoimt his experience and available information about the system under study (Biuton and Nickless, 1987). In Fig. 8.1a, the first simplex is defined by the A, B and C vertexes. Performing experiments at the conditions indicated by these vertexes and comparing the results, we verify that they correspond, respectively, to the worst, second worst and the best of the three observed responses. You can easily verify this by observing the location of the simplex relative to the contour curves of the response surface. This classification is necessary so that we can define the location of the second simplex, done according to rule 2. 

The worst response of the new simplex (BCD) occurs at the B vertex, whose reflection results in the CDE simplex (Fig. 8.1b). Doing this several times, we get a sort of zig-zag displacement with its resultant almost perpendicular to the contour curves of the response surface, approximately corresponding to the path of maximum inchnation (Fig. 8.1c). 

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To permit a more rapid convergence towards an optimum, several modified simplex methods have been suggested.[2] By these modified methods, the step-length of the next move is adjusted depending on the degree of improvement. If a new vertex should give a considerable improvement, it is rather natural to try to move further in that direction. One such modified simplex method is given in detail below, after a presentation of the basic simplex method. 

In the Basic simplex method, each vertex is at equal distance to other vertices, the simplex is regular. In two dimensions it is an equilateral triangle in three dimensions it is a regular tetrahedron. Each move will therefore have a fixed step-length due to this constraint. The method is defined by a set of rules ... 

J. Example Optimization of a Friedel-Crafts alkylation by the Basic simplex method... 

The method is described by a set of rules (as for the Basic simplex method). The principles are illustrated by a simplex in two variables. The formulae for computations are gereral and can be used for any number of variables. Vector notations are used to describe the method. It is assumed that a maximum response is desired. If a minimum is to be found, all relations greater than (>) and less than (<) should be reversed in the following rules. 

The Basic simplex method is good when the experimental domain is not too large. 

In the basic simplex procedure, proposed by Spendley et al. (98), the first three experiments are performed according to the conditions of the initial simplex, called BNW (Figure 2.12). B, N, and W correspond to the vertices with the best, next-to-best, and worst responses, respectively. The best response is usually either the highest or the lowest, depending on what is the most desired situation. The size of the initial simplex is arbitrarily chosen by the analyst. B, N, and W can be represented by the vectors b, n, and w, that is, b = xii,X2b, n = [xi ,X2 ], and w = xi ,X2 . Depending on the obtained results, the next experiment will be selected. 

The basic simplex procedure is further described by four rules (9,10, 98, 99). 

In Figure 2.13, an example is given of the basic simplex procedure. Consider the imaginary response surface of a method, representing the response as a function of two factors (xi and X2) and shown as contour plot (dotted lines). Suppose the highest response value is considered to be the optimum. 

In the basic simplex method, the simplex thus can only be reflected to obtain the next experiment, and the simplex size remains the same throughout the procedure. In the modified simplex method, suggested by Nelder and Mead (100), the simplex can be reflected, expanded, or contracted to define the next experiment. Thus, in case the simplex is expanded or contracted, the simplex size changes. More information about the simplex procedures can be found in References 7,9,10, and 98-102. 

The coordinates of the D-1 designs for up to seven factors can be obtained from Table 6.19. The initial simplexes from which the designs are generated are built sequentially, adding to the previous simplex the row under the appropriate dashed hne, and filling the blanks in the last column with zeroes. For example, the basic simplex for = 5 is obtained by adding rows 5, 8 and 12 to the first three rows. This results in the 6x5 matrix... 

To construct Doehlert designs of type D-3, one starts with a simplex with a vertex on the origin and totally located in the first quadrant of the coordinate system, having the bisector plane as a plane of symmetry. By the usual subtraction process, this simplex generates symmetrical designs where aU factors have seven levels. For k factors, the basic simplex has the general form... 

Fig. 8.3 shows what winds up happening when the basic simplex comes sufficiently close to the pursued value. Having arrived in the neighborhood of the maximum, which is the desired value in this example, the simplex starts to describe a circular movement around the highest observed response (point A, in Fig. 8.3), and is not able to move from there. At this point there is nothing more for us to do, since the basic simplex cannot reduce its size. The optimization process should be stopped, and the precision with which the optimized conditions are determined depends on the size and location of the initial simplex. In our example, the maximum response is a little more than 86. The maximum value reached by the simplex is quite close about 85.3, at the experimental conditions defined by the A vertex, s50 and c s247. 

Fig. 8.3. Circular movement of the basic simplex in the vicinity of the maximum. Vertex A is retained in all the simplexes.

In the modified algorithm (Nelder and Mead, 1965), the simplex can change its size and form, and consequently adapt itself more efficiently to the response surface. This flexibility permits a more precise determination of the optimum point, because the simplex can shrink in its proximity. Besides this desirable characteristic, the modified method, compared to the basic simplex, can reduce the number of runs necessary to find the optimum, because it can stretch itself when it is far from the desired point, usually on a planar portion of the response surface. For this reason it approaches the experimental region of interest more rapidly. 

With vectorial Equations (8.1) and (8.2) we can determine the coordinates of all the points spanned by the basic simplex. 

In Figs. 8.5 and 8.6, we use the same response surface as the one in Figs. 8.1 and 8.2 to illustrate the application of the modified simplex. The corresponding numerical values of concentration and time for the various vertexes are given in Table 8.1. Note that the initial ABC simplex is the same as in the example of the basic simplex, permitting us to compare the efficiencies of the two algorithms. 

Optimization of Electrolyte Properties by Simplex Exemplified for Conductivity of Lithium Battery Electrolytes, Fig. 1 The basic simplex method is based on simple reflections (R) of the simplex (shown in the lower right). The optimization begins at starting simplex (Xi,Yi,ZilX2,Y2,Z2lX3,Y3,Z3), soUd line. A new set of... 

The basic simplex optimization method, first described by Spendley and co-workers in 1962 [ 11 ], is a sequential search technique that is based on the principle of stepwise movement toward the set goal with simultaneous change of several variables. Nelder and Mead [12] presented their modified simplex method, introducing the concepts of contraction and expansion, resulting in a variable size simplex which is more convenient for chromatography optimization. 

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