Ordering fuzzy quantities is a challenging problem in fuzzy sets theory that has attracted the interest of many researchers. Despite the multiple indices introduced for this purpose and due to the fact that fuzzy quantities do not have a natural order, there is still a chance to provide a new approach for ranking this type of quantities from the acceptability and foundation point of view. This paper aims at developing a new approach to ranking fuzzy numbers (FNs), fuzzy rank acceptability analysis (FRAA), which not only implements a ranking of the FNs, but also provides a degree of confidence for all ranks. Additionally, the FRAA can be efficiently implemented by using different fuzzy preference relations including both transitive and intransitive ones. Properties of FRAA ranking, their dependence on the fuzzy preference relations, and correspondence with the basic axioms for ranking FNs are analyzed. Finally, a comparison of the FRAA ranks to ranks from other methods is analyzed along with a discussion of the advantages of FRAA ranking.

中文翻译：

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模糊秩可接受性分析：对模糊数排序的置信度

模糊量的排序是模糊集理论中的一个具有挑战性的问题，引起了许多研究人员的兴趣。尽管为此目的引入了多个指标，并且由于模糊量没有自然顺序，但仍然有机会从可接受性和基础的角度提供一种新的方法来对此类量进行排序。本文旨在开发一种对模糊数 (FN) 进行排序的新方法，即模糊等级可接受性分析 (FRAA)，它不仅实现了对 FN 的排序，而且还为所有等级提供了一定程度的置信度。此外，通过使用不同的模糊偏好关系，包括传递性和非传递性的，可以有效地实现 FRAA。FRAA 排序的属性，它们对模糊偏好关系的依赖，并分析了与用于对 FN 进行排序的基本公理的对应关系。最后，分析了 FRAA 排名与其他方法排名的比较，并讨论了 FRAA 排名的优势。