Simplex regular plan

The simplex regular plan can be introduced here with the following example a scientist wants to experimentally obtain the displacement of a y variable towards an optimal value for a y = f (xj,X2) dependence. When the analytical expression y = f (xj,X2) is known, the problem becomes insignificant, and then experiments are not necessary. Figure 5.11A shows that this displacement follows the way of the greatest slope. In the actual case, when the function f(xj,X2) is unknown, before starting the research, three questions require an answer (i) How do we select the starting point (ii) Which experimental and calculation procedure do we use to select the direction and position of a new point of the displacement (iii) When do we stop the displacement ... 

For k factors, the number of experiments required by the simplex regular matrix is N = k-rl. So, the class of saturated plans contains the simplex regular plan where the number of experiments and the number of the unknowns coefficients are the same. For the process characterization in this example, we can only use the relationships of the linear regression. Concerning the simplex regular matrix... 

For practical use, the simplex regular plan must be drafted and computed before starting the experiment. For k process factors, this matrix plan contains k columns and k-i-1 lines in the case of k = 6 the matrix (5.151) gives the following levels of the factors ... 

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. 

To conclude this section, it is important to mention that the method of simplex regular plan is an open method. So, during its evolution, we can produce and add additional factors. This process can thus result in a transformation from a simplex regular plan with k columns and k-tl lines to a superior level with k-tl columns and k-i-2 lines. The concrete case described in the next section shows how we use this method and how we introduce a new factor into a previously established plan. 

Table 5.34 Simplex regular plan with values of natural factors (second step of example 5.5.6.1).

We define a regular simplex plan as an assembly of k+1 equidistant points for k= 1, the simplex is a segment for k= 2, it is a triangle for k= 3, we are faced with a regular tetrahedron, etc. Each simplex has a geometric centre placed at one point. 

If the dimensionless factors of an investigated process are distributed in a planning matrix (5.138) where Xj values are obtained using relation (5.139), then we can prove that the points of the matrix are organized as a regular simplex. Relation (5.140) corresponds to the distance from a point to its opposite face. 

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