Multidimensional Fuzzy Sets

Problems can arise in decision making when a different number of evaluations for the different criteria or alternatives may be available. This is the case, for instance, when for some reason, one or more experts refrain from evaluating certain criteria for an alternative. In these situations, approaches using $n$ -dimensional fuzzy sets and hesitant fuzzy sets are not appropriate. This is because the first approach works only with a fixed dimension, whereas the second modifies the information, fusing equal evaluations given by different experts for the same pair of alternatives/criteria or adjusting the elements so that an order defined on hesitant fuzzy sets can be used. As such, in addition to being a natural extension of $n$ -dimensional fuzzy sets, multidimensional fuzzy sets contemplate the aforementioned situation without modifying their elements. In this article, we introduce the concept of multidimensional fuzzy sets as a generalization of the $n$ -dimensional fuzzy sets in which the elements can have different dimensions. We begin by presenting a way to generate a family of partial orders and conditions under which these sets have a lattice structure. Moreover, the concepts of admissible orders on multidimensional fuzzy sets and $m$ -ary multidimensional aggregation functions with respect to an admissible order are defined and studied. Finally, we provide two multicriteria group decision-making methods based on these $m$ -ary multidimensional aggregation functions.