On various approaches to normalization of interval and fuzzy weights

The paper deals with the problem of fuzzification of the procedure of normalization of weights. First, the existing methods for normalization of interval and fuzzy weights are reviewed. Second, we study the problem of normalization of a fuzzy vector of weights that expresses the joint possibility distribution of initial weights. We show that a correct way is to apply the extension principle proposed by Zadeh, since the result of such normalization is the fuzzy vector of normalized weights that expresses the true joint possibility distribution of normalized weights. Further, we establish some properties of this approach to normalization that are important from the point of view of real applications. Finally, since an n-tuple of non-interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind, we investigate normalization of such kind of fuzzy vectors of weights according to the extension principle. We show that from the point of view of the way of modelling uncertain normalized weights, the result of this approach can be directly compared only with the result of normalization proposed by Wang and Elhag (2006) [35]. We find out that it is not sufficient to express the result of normalization only by an n-tuple of normalized interval or fuzzy weights together with the constraint that the sum of the weights is equal to 1, since it can cause a false increase of uncertainty in the model. This fact is illustrated by an example.