Simplifying the Search for Effective Ranking of Fuzzy Numbers

We prove that a ranking index which generates an order on some subset of fuzzy numbers can be reduced to a ranking index with a simpler form and which generates an equivalent order. This way, the finding of ranking indices generating orders with desirable properties can be essentially simplified such that we need to search only from additive or scale invariant ranking indices. We determine exactly, making an abstraction of equivalent orders, the class of ranking indices on trapezoidal fuzzy numbers which generate orders satisfying all the basic requirements. Finally, ranking indices used to rank trapezoidal fuzzy numbers are extended to ranking indices used to rank arbitrary fuzzy numbers so that the desired properties are preserved.