Explicit analytical formulae of ranking indices without the requirement of multiplicative compatibility.

The ranking of fuzzy numbers is often given by a special class of utility functions, better known as ranking indices of the first kind. Recently, we have proved that the effective ranking of fuzzy numbers from a set containing all real numbers can be done by involving a very particular class of ranking indices. In this way, the searching of effective ranking methods is essentially simplified. In this paper, firstly we extend this result in a sense which allows us to find explicit formulae for ranking indices defined on other important subsets of fuzzy numbers, as for example the set of positive trapezoidal fuzzy numbers or the set of positive triangular fuzzy numbers. Another novelty is that, reducing all those ranking indices that generate the same ordering to a special member of their class, under some reasonable regularity assumptions, we find all the orderings on trapezoidal fuzzy numbers, or positive trapezoidal fuzzy numbers, respectively, which satisfy all the basic requirements of Wang and Kerre, except for the compatibility with scalar multiplication. This special member of the class is what we call a basic ranking index, that is, a ranking index which assigns to a fuzzy number a real number that belongs to its support. Actually, these basic ranking indices can be extended to linear functionals, so their form is very simple and ready to be used in applications. Similar results are obtained for the set of triangular fuzzy numbers and positive triangular fuzzy numbers, respectively. The key element in all these theoretical results, is that under some reasonable regularity properties, we can reduce a ranking index to this so called basic ranking index, which has a simpler form but generates the same order, that is, a binary relation with the same graph. As a first application, we review several ranking approaches got from ranking indices which are additive functions with respect to addition of fuzzy numbers and check their properties from the list of Wang and Kerre. Then, we do the same for a group of ranking indices none of them being additive functions with respect to the addition of fuzzy numbers. We can always do that, as long as we reduce a nonadditive ranking index to a basic ranking index, with both ranking indices generating the same ordering. For the newly obtained ranking index, we will just need to study additivity or scale invariance, depending on which type of property from the list of Wang and Kerre are we interested in. An interesting special case of nonadditive ranking index that cannot be reduced to a basic ranking index is the so called signed distance. We slightly modify this ranking index, so that it can be reduced to a basic ranking index. In this way, we never get contradictory results for fuzzy numbers with disjoint supports. Lastly, we propose a method to extend the ordering of positive trapezoidal fuzzy numbers to the ordering of arbitrary positive fuzzy numbers.