**1. Introduction**

The ventilation system is one of the most important technologies to ensure the safety of mines [1]. In the process of mining, it is necessary to provide enough fresh air and exclude harmful gases, heat and dust [2]. Then, a good working environment can be created to guarantee the health and safety of underground workers. Therefore, choosing an applicable mine ventilation system is essential and important for mines. Since there is much ambiguity and uncertainty in the evaluation process, the selection of mine ventilation systems can be deemed as a fuzzy decision making problem.

In the process of decision making, experts or decision makers (DMs) may prefer to do comparisons among each pair of systems or construct a preference matrix when expressing their opinions [3,4]. On the other hand, because of the complexity of alternatives and the fuzziness of human cognitions, many people may tend to give preference information in the form of language phrases, such as "good", "bad" and so on [5–7]. Thereafter, the decision making problems based on linguistic preference relations (LPRs) have attracted extensive attention [8–10]. However, there is a hypothesis that the

membership degree of each element in LPRs is a certain number "one". It is unable to accurately depict the professionals' supporting or hesitant degrees of linguistic assessment information.

Accordingly, Rodriguez et al. [11] put forward the concept of hesitant fuzzy linguistic term set (HFLTS) to express the experts' hesitation or inconsistency. HFLTS is an orderly limited subset of linguistic terms. Different varieties of aggregation operators [12–14], measures [15,16] and decision making approaches [17–19] based on HFLTS and its extensions were proposed. For instance, Liu et al. [20] defined the distance measures for HFLTS to deal with hesitant fuzzy linguistic multi-criteria decision making problems; Adem et al. [21] proposed an integrated model using SWOT analysis and HFLTS for evaluating occupational safety risks in the life cycle of wind turbines. Besides, Zhu and Xu [22] came up with the concept of hesitant fuzzy linguistic preference relations (HFLPRs), where the basic elements are in the form of HFLTS.

Subsequently, numerous researchers had grea<sup>t</sup> interest in studying preference matrices under hesitant fuzzy linguistic conditions. Zhang and Wu [23] introduced the multiplicative consistency of HFLPRs based on distance measures. Wang and Xu [24] defined the additive and weak consistency of extended HFLPRs on the basis of graph theory. Wu and Xu [25] discussed the consistency and consensus of HFLPRs in a group decision environment. Gou et al. [26] proposed the compatibility measures and weights determination approach for HFLPRs, and then applied them in selecting a desirable aspect in the medical and health system reform process. Li et al. [27] recommended an approach of obtaining the interval consistency degree of HFLPRs. Xu et al. [28] constructed a group decision support model for HFLPRs to reach consistency and consensus.

Nevertheless, HFLTS cannot reflect the membership degree of an element that belongs to a specific concept [29], such as a certain ventilation system in this paper. It is only a collection of several linguistic evaluation values, and it has strong subjectivity and fuzziness [30]. In order to overcome the inherent defects of linguistic variables and HFLTS, hesitant fuzzy linguistic sets (HFLSs) were introduced by Lin et al. [31]. They can describe hesitant degrees of DMs with some membership degrees based on a given linguistic term. Compared with uncertain linguistic variables, they have the edge on describing the fuzziness [29]. HFLSs combine linguistic term sets with hesitant fuzzy sets (HFSs), which include both the quantitative and qualitative evaluation information [32]. Each element in the HFLSs can be called a hesitant fuzzy linguistic number (HFLN). For instance, half of the specialists in Group A think that *vs*1 is a good ventilation system, and 80 percent in Group B think so. In this case, it can be expressed with a HFLN < *good*, {0.5, 0.8} >.

The motivations of this paper are mainly two-fold. (1) The mine ventilation systems selection context requires dealing with fuzzy evaluation information and building appropriate decision making models. Hesitant fuzzy linguistic numbers (HFLNs) have advantages in describing the fuzziness and hesitancy of experts [29]. Moreover, preference relations are among the most powerful tools to select the best system. (2) Currently, researches on HFLNs are relatively insufficient compared with other fuzzy sets. Wang et al. [29] developed a decision making method based on the Hausdorff distance of HFLNs. In addition, Wang et al. [30] put forward the concept of interval-valued HFLNs to deal with complex decision making issues. Yet, there are still some limitations with existent operational laws and comparison methods of HFLNs [29,31].

Taking the aforementioned motivations into account, this paper concentrates on selecting the optimal mine ventilation system under a hesitant linguistic environment.

The novelty and contributions of this paper are listed as follows.


HLPRs and a consistency-improving model is introduced to ge<sup>t</sup> acceptable consistency. Besides, a likelihood-based method is adopted to obtain the final ranking result.

(3) The proposed method is applied in the engineering field of choosing appropriate mine ventilation systems. Thereafter, an in-depth comparison analysis is conducted to demonstrate the validity and merits of the presented method.

The remainder of this paper is arranged as follows: Introductory knowledge about HFLSs and preferences relations are briefly reviewed in Section 2. Section 3 proposes new operations and comparison method of HFLNs. A consistency index is put forward for checking the consistency level of HLPRs in Section 4. A consistency-improving process may be carried out when a HLPR's consistency is unacceptable. And a likelihood-based approach is presented to ge<sup>t</sup> the ranking results subsequently. Section 5 illustrates an example of ventilation systems selection and makes a comparative analysis to express the effectiveness of the proposed method. Necessary discussions and brief comments are informed in Section 6.
