3.5.4 Concluding Remarks

All the three ranking methods use a procedure, which has a common general structure, comprising two steps. In the first step an intermediate aggregated relational structure is created from the individual preference data. In the second step, the aggregated relational structure is used to determine the final rank–order structure.

The input to the second step is a fuzzy relation for both the fuzzy methods. Naturally, these methods use fuzzy logic in the second step and the results are also in the fuzzy form. But the final structure they are looking for is not the same. Fuzzy method–2 assumes a total –linear–order and determines the final output membership function according to this assumption. If the data cannot be fitted to such a model, then we shall obtain low membership function values. In Fuzzy method–1, the precondition is a more general relational structure in which equivalent non–dominated layers are successively determined. Moreover, it is possible that in the set of alternatives (or in a part of it) such layers do not exit. In this case, we may find alternatives not belonging to a core, but with membership values close to the maximum. This is a sign of instability of the results ( implied by the input data).

The method Electre constructs a non–fuzzy relationship in the first step (although in a sub–step one can find fuzzy elements), and in this respect it differs from both the fuzzy methods. But in the second step, the same approach is used as in Fuzzy method–1 In this respect, the only difference is that the concepts of classical logic are used. Therefore, the results issued by Electre are in a non–fuzzy form. The strength and stability of the relational structure can be characterized by parameters controlling the ELECTRE procedure.

The three ranking methods described above can be applied to the same problems, and their results can reflect the same phenomena from different aspects, which may provide a deeper understanding of the information represented in the data than any single method.