On the Conditions for Total Orderings in Lexicographic Methods to Rank Fuzzy Numbers

Lexicographic methods to rank fuzzy numbers present the advantages of simplicity, consistency with human intuition, and power of discrimination. In this paper, we tackle the problem of finding the conditions for these methods to produce a rank in specific steps. Our main results are twofold. First, we prove that a necessary and sufficient condition for a ranking function to be a total order is that this function is either injective, surjective, or bijective. Second, we provide further insight into the required steps for a lexicographic order to rank same-type and different-type fuzzy numbers. A counterexample refutes a conjecture in the literature about the maximum number of steps needed to rank different-type fuzzy numbers.