## 1 Introduction  
In Group decision making (GDM), various decision makers (DMs or experts) would be employed to express their preferences via pairwise comparison over decision alternatives. Two classical preference relations consist of fuzzy preference relations (FPRs) [1; 2] and multiplicative preference relations (MPRs) [3; 4], in which the elements are described by exact numerical values. However, due to the uncertainty and complex information granularity, DMs may have difficulty in providing pairwise comparison judgments on alternatives with crisp numerical values. To address this issue, the intuitionistic fuzzy preference relation (IFPR) [5; 6], interval-valued fuzzy preference relation (IVFPR) [7; 8] and interval-valued intuitionistic fuzzy (IVIF) preference relation (IVIFPR) [9; 10] appear one after another. Particularly, the elements in an IVIFPR consist of IVIF values (IVFVs), where both the membership degree and non-membership degree are intervals. Therefore, IVIFPR can flexibly seize such uncertainty and vagueness information in decision making. In recent decades, IVIFPR has been widely used to solve different GDM problems, such as the supplier selection [11], the virtual enterprise partner selection [12], the transport service provider selection [13], the air traffic protection aircraft [14] and so on. The consistency degree of comparison matrices directly affects the final decision result. For GDM problems, the consistent IVIFPRs provided by DMs refer to certain transitivity property, which can ensure that DMs' judgments contain no contradiction in some sense. Therefore, it is natural that the consistency of IVIFPRs should be measured and improved in GDM problems. To date, a host of studies have been devoted to discussing this issue. For instance, [PERSON] and [PERSON] [15] first proposed the concept of the consistent interval-valued intuitionistic judgment matrix. Subsequently, [PERSON] and [PERSON] [16; 17] introduced some definitions on multiplicative and additive consistency in incomplete IVIFPR and determined the missing elements in an IVIFPR while only knowing its off-diagonal elements. Inspired by the multiplicative transitivity of an IVIFPR, [PERSON] et al. [18] proposed a multiplicative consistency concept to an IVIFPR, which was applied to adjust or repair an inconsistent IVIFPR through using some iterative algorithms. These achievements directly investigated the multiplicative transitivity property of an IVIFPR. In contrast, [PERSON] et al. [12] indirectly examined the multiplicative consistency of an IVIFPR by inducing two special IFRP matrices from an IVIFPRs. Subsequently, with the same idea, [PERSON] et al. [9; 10] defined the additive consistency of an IVIFPR by generating two special IVFPRs and extracting two consistent IFFR from an IVIFPR, respectively. [PERSON] et al. [11] put forward a new definition on the additive consistency of an IVIFPR by dividing an IVIFPR into four preference relations. [PERSON] et al. [19] built 0-1 mixed programming models to judge the multiplicative consistency of IVIFPRs, and developed a multiplicative consistency and consensus based algorithm to GDM with IVIFPRs. [PERSON] et al. [20] used statistical approach and presented a sensitivity analysis model to discuss the result for consistency evaluation of decision making. Since the priority weights can be utilized to rank alternatives, it is an important issue to determine the priority weights for GDM problems based on preference relations [21]. Due to the complex operations on IVIFVs, research paper on determining priority weights from IVIFPRs are very scarce. For example, [PERSON] and [PERSON] [15] developed an approach to GDM with IVIFPR by the ordered weighted and hybrid aggregation operators. [PERSON] and [PERSON] [22] proposed a similarity measure between IFPRs and extended it to IVIFPRs. [PERSON] and [PERSON] [23] introduced a new attitudinal expected score function for interval-valued intuitionistic fuzzy numbers. [PERSON] [24] presented a geometric approach for ranking IVIFVs. [PERSON] et al.