Interval-valued intuitionistic fuzzy sets and vague sets are two sets that are commonly used in expressing vague and uncertain information. Despite their commonality, these two sets are characterized by different memberships. Interval-valued intuitionistic fuzzy sets are characterized by membership and non-membership with the sum of these memberships being less or equal than one. On the other hand, vague sets are characterized by either truth or false memberships with no clear-cut values of their sum. However, these two sets have shown a great extent of superiority in dealing with uncertain and imprecise information despite differences in the memberships. Therefore, it is a worthy effort to develop a new generic set by combining these two sets: interval-valued intuitionistic fuzzy sets and vague sets. In this paper, the notion of interval-valued intuitionistic fuzzy vague sets (IVIFVS) is proposed where membership and non-membership of interval-valued intuitionistic fuzzy sets are combined with truth membership and false membership of vague sets. To complement this new definition, several mathematical propositions, algebraic relations and arithmetic operations of IVIFVS are presented. Moreover, a new linguistic variable of IVIFVS with five linguistic scales that are normally used in solving multi-criteria decision-making problems is developed. Some examples are provided to illustrate the conceptual definitions and linguistic variables of the new sets.

区间直觉模糊集和模糊集是表达模糊和不确定信息时常用的两个集。尽管它们具有共同点，但这两个集合的特征在于成员不同。区间值直觉模糊集的特征在于隶属度和非隶属度，这些隶属度的总和小于或等于1。另一方面，模糊集的特征是真隶属关系或虚假隶属关系，其总和没有明确的值。但是，尽管成员不同，但这两套方法在处理不确定性和不精确信息方面显示出很大的优势。因此，通过结合这两个集合（区间值直觉模糊集合和模糊集合）来开发新的泛型集合是值得的。在本文中，提出了区间值直觉模糊模糊集（IVIFVS）的概念，其中区间值直觉模糊集的隶属度和非隶属度与模糊集的真隶属度和虚假隶属度相结合。为了补充这个新定义，提出了IVIFVS的几个数学命题，代数关系和算术运算。此外，还开发了一种新的IVIFVS语言变量，该变量具有五个语言等级，通常用于解决多准则决策问题。提供一些示例来说明新集合的概念定义和语言变量。为了补充这个新定义，提出了IVIFVS的几个数学命题，代数关系和算术运算。此外，还开发了具有五个语言等级的IVIFVS的新语言变量，通常用于解决多准则决策问题。提供一些示例来说明新集合的概念定义和语言变量。为了补充这个新定义，提出了IVIFVS的几个数学命题，代数关系和算术运算。此外，还开发了具有五个语言等级的IVIFVS的新语言变量，通常用于解决多准则决策问题。提供一些示例来说明新集合的概念定义和语言变量。