## 1 Introduction
[17] studied a neuro-fuzzy architecture employing CFSs; [PERSON] and [PERSON] [18] reviewed the CFSs; [PERSON] et al. [19] proposed some applications of CFSs; [PERSON] et al. [20] formulated distance measures between IVCFSs; [PERSON] et al. [21] defined the join and meet operations for IVCFSs. According to the definition of an FS, it only discusses the degree of membership \(\hat{\mathfrak{m}}\), but it also covers the degree of non-membership \(\hat{\mathfrak{n}}\), which is the complement of the degree of membership with respect to unit interval, i.e., \(\hat{\mathfrak{n}}=1-\hat{\mathfrak{m}}\). This hidden feature of FSs led to the definition of the intuitionistic fuzzy set (IFS), proposed by [PERSON] [22]. An IFS associates two mappings to each of its elements, which are called the degree of membership and the degree of non-membership. Both of these mappings take on values from \([0,\,1]\) interval given that their sum also lies within the range of this interval. The idea of the intuitionistic fuzzy relation (IFR) was concocted by [PERSON] et al. [23]. In 1999, [PERSON] [24] expressed the degree of membership and the degree of non-membership of an IFS in the form of intervals and, hence, developed a new concept called the interval-valued intuitionistic fuzzy set (IVIFS). Complex valued mappings were brought in to IFS theory by [PERSON] et al. [25] in 2012, who introduced the notion of the complex intuitionistic fuzzy set (CIFS). [PERSON] and [PERSON] [26] put forward the interval-valued complex intuitionistic fuzzy relation (IVCIFS). [PERSON] [27] used IFSs for multiattribute decision-making (MD) models and methods; [PERSON] et al. [28] applied IFSs in medical diagnosis; [PERSON] and [PERSON] [29] applied IFSs to pattern recognition; [PERSON] et al. [30] compared IVFSs, IFSs and bipolar-valued FSs; [PERSON] [31] used Hausdorff metric for finding the distances between IVFSs and IFSs; [PERSON] et al. [32; 33; 34; 35] applied complex relations to analysis of economic relationships. [PERSON] et al. [36] studied complex intuitionistic fuzzy classes; [PERSON] and [PERSON] [37] proposed a new distance measure of IVIFSs and applied it in decision making; [PERSON] and [PERSON] [38]devised the correlation of IVIFSs; [PERSON] and [PERSON] [39] proposed a method of ranking IVIFSs. [PERSON] et al. [40] proposed the application of fuzzy logic in assessing the security control for organizations; [PERSON] et al. [41, 42] also used fuzzy theory to rank and prioritize the security controls for cloud wireless sensor networks and computing networks; [PERSON] et al. [43] used fuzzy multiple-attribute decision-making techniques for selecting the best control system. In this paper, the Cartesian product (CP) of two IVCIFSs is introduced. Furthermore, the innovative conception of an interval-valued complex intuitionistic fuzzy relation (IVCIFR) is defined by using the concept of the CP of IVCIFSs. Moreover, the types of IVCIFRs have been defined, including the interval-valued complex intuitionistic converse fuzzy relation, interval-valued complex intuitionistic equivalence fuzzy relation, interval-valued complex intuitionistic partial order fuzzy relation, interval-valued complex intuitionistic composite fuzzy relation and many more. Every definition is held by the examples. In addition, some results have been proved for the type of IVCIFRs. Besides these, the Hasse diagrams for interval-valued complex intuitionistic partial order fuzzy sets and relations have also been presented. Furthermore, the ideas related to a Hasse diagram: maximum element, minimum element, maximal element, minimal element, supremum, infimum, upper and lower bounds are discussed. The novel concepts introduced in this study, i.e., IVCIFSs and IVCIFRs are superior to the pre-existing frameworks of FSs, IFSs, IVFSs, IVIFSs, CFSs, CIFSs and IVCIFSs. Since IVCIFRs analyze the relationships among the IVCIFSs, so they are composed of the degrees of membership and non-membership, which are in the form of intervals with complex numbers.