## Bwpcnh 2 I n lK 2p

Evidently, Equation 24.23 becomes identical to Equation 24.13, corresponding to the parallel system, when the number of levels E is one.

The following step is to carry out the comparison between the expressions for the parallel and the hierarchical systems and to try to obtain an analytic expression that allows the decision of the better solution for a given problem. However, if the MF of the parallel case is taken as a reference, one would have one equation and 4E variables, which would determine infinite solutions.

A different approach to compare both systems would be the following:

1. The parallel solution is determined and its MF is calculated.

2. A hierarchical system is designed and its MF is calculated.

3. Both results are compared. If one is interested in a hierarchical solution and its MF is bigger than the one for the parallel solution, the hierarchical parameters (a, b, and P) can be adjusted and the process repeated until the MF is smaller.

However, it is possible to obtain an analytic expression if what is known is the value of the MF of the parallel system and the values of the parameters of E - 1 levels of the hierarchical system of E levels. In this case, imposing the condition that, for example, the MF in the hierarchical system is smaller than the one in the parallel system, the following expression is obtained to calculate the values of the parameters of the last level:

NE-aE + (Ne - l)K2-P_ N.b r , , N.-a. + (n. - l)-K2 P E E \ E > 2 E. .[N-a + (N - 1)-K2 ]] ' ' .- 2 ' (24.24)

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