The existence of fuzzy priorities is here grounded on a recent proposal for fuzzy-linguistic preference structures (Franco, Knowledge-Based Systems 70 (2014) 203), where the outcome of the decision ordering-process is a fuzzy weight for every decision object. Hence, a procedure is needed for ranking such fuzzy priorities. Here a ranking methodology is suggested based on the core and support of fuzzy weights, which represent the priority intensity for decision objects. In this way, the available evidence is evaluated according to its relative importance, allowing the inclusion of human-expert knowledge, where an automatic search is performed by computing the robustness/stability of the solution, identifying some confidence bounds for the estimated degrees of importance. As a result, building on the linguistic preference statements of experts, the system recommends informative and robust ranking-solutions.