## 1 Introduction  
In a group decision making (GDM) situation, the decision makers (DMs) are usually required to select the desirable alternative(s) from a collection of alternatives. To cope with this problem, DMs would compare alternatives with each other and provide the preference information, and a judgement matrix can be constructed [1; 2; 3]. In order to model DMs' knowledge and preferences, preference relations have been introduced. To characterize fuzziness and uncertainty, some kinds of extended preference relations have been introduced, including fuzzy preference relation (FPR) [4; 5; 6; 7; 8], multiplicative preference relation (MPR) [9; 10; 11; 12] and linguistic preference relation (LPR) [13; 14]. The experts describe their preference information with a 0-1 scale crisp numbers in FPR, and they utilize a 1-9 scale to express their preference information in MPR [15]. It is noted that the elements in MPRs are crisp values. However, considering the fuzziness and hesitation involved in practical decision-making problems, it may be difficult for DMs to express their evaluated information with crisp values. To describe the imprecision, the interval multiplicative preference relation [16] and the intuitionistic multiplicative preference relation (IMPR) [17] are introduced to express their decision-making preference information. [PERSON] et al. [18] first defined the intuitionistic multiplicative preference relation (IMPR) and developedsome IMPRs information aggregation techniques. [PERSON] and [PERSON] [18] introduced the concepts of hesitant fuzzy preference relation (HFPR) and hesitant multiplicative preference relation (HMPR) and studied their properties, which are followed by the construction of some methods of group decision making (GDM). For various forms of preference relations, the most two important issues are consistency analysis and consistency improvement [19]. [PERSON] et al. [20] developed an approach to check the inconsistency and weak transitivity for the FPR and to repair its inconsistency for reaching weak transitivity. [PERSON] et al. [21] designed a method to construct consistent FPRs from a set of original preference data. For a given MPR, [PERSON] and [PERSON] [16] proposed a convergent model to improve its consistency. With the help of the Abelian linearly ordered group, [PERSON] and [PERSON] [22] established a general improving consistency and reaching consensus methods for different types of preference relations. By using the order consistency and multiplicative consistency, [PERSON] et al. [23] proposed two new approaches for GDM with IFPRs to produce the normalized intuitionistic fuzzy weights for alternatives. [PERSON] [24] proposed some linear programming models for deriving intuitionistic fuzzy weights. For the unbalanced LPRs, [PERSON] et al. [25] investigated an optimization model to increase the consistency level. [PERSON] et al. [26] developed an iterative algorithm to adjust the additive consistency of IFLPRs and derive the intuitionistic fuzzy weights for IFLPRs. Based on \(\beta\)-normalization, [PERSON] et al. [27] utilized the optimized parameter to develop a novel approach for inconsistent HFPRs. Under the hesitant fuzzy preference information environment, [PERSON] et al. [28] constructed a decision support model to derive the most desirable alternative. Similar to MPRs, studying the HMPR is an important research topic. However, there are few techniques in the existing literature have been done about it. [PERSON] and [PERSON] [18] directly used the proposed operators to aggregate the HMPRs information. However, it is generally known that the unacceptable consistency preference relations easily lead to inconsistent conclusions. Therefore, the decision-making results that obtained by the method in [PERSON] and [PERSON] [18] may be unreasonable. Based on the \(\beta\)-normalization principle, [PERSON] and [PERSON] [28] investigated a new decision-making model to generate the interval weights of alternatives from HMPRs.