**1. Introduction**

In many multi-criteria decision making (MCDM) problems, because of the incomplete information and the complexity of the decision-making environment, crisp numbers cannot describe the relevant decision information. Thus Zadeh [1] proposed the fuzzy set (FS) *A* = { -*xj*, *<sup>μ</sup>A*-*xj*..HH*xj* ∈ *X* } -0 ≤ *<sup>μ</sup>A*-*xj*. ≤ 1. on *X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*}, where *<sup>μ</sup>A*-*xj*. is the membership degree of *xj* ∈ *X*. Since it was put forward, many scholars have generalized it. For example, Atanassov [2,3] introduced the concepts of the intuitionistic fuzzy set (IFS) and the interval-valued intuitionistic fuzzy set (IVIFS), and Torra [4] proposed the hesitant fuzzy set (HFS). In the past few years, the FS and its extensions have been applied in many fields, such as supplier selection, pattern recognition, and medical diagnosis. As the FS and its extensions mentioned above use crisp numbers to express decision information, they cannot express qualitative evaluation information. For instance, when one expert evaluates the performance of a company, he/she thinks that the performance of the company is very good. Because the evaluation expression is consistent with the human's cognitive process, it is suitable to express this in a linguistic term set (LTS). To describe the relevant information, Zadeh [5–7] proposed the LTS to express the relevant information. A general discrete LTS of seven terms can be represented as *S* = {*<sup>s</sup>*0 : *very poor*, *s*1 : *poor*,*s*<sup>2</sup> : *slightly poor*, *s*3 : *f air*, *s*4 : *slightly good*, *s*5 : *good*, *s*6 : *very good*}. Then, the expert's evaluation about the performance of the company can be represented as {*<sup>s</sup>*6}. However, due to the uncertainty of the problem in the decision making process, the decision makers cannot express their preferences using only one membership degree of a LTS. In order to express the decision makers' hesitation about the decision problem, Rodríguez et al. [8] proposed the hesitant fuzzy linguistic term set (HFLTS), which is based on LTS and HFS. The HFLTS makes the representation of the decision information more flexible. Since the HFLTS was proposed, a number of relevant studies and their applications [9–22] have been conducted. For example, Liu et al. [16] presented the fuzzy envelope for HFLTSs and applied it to choose the best alternative. Xu et al. [17] presented the hesitant fuzzy linguistic ordered weighted distance operator, and applied it to plan the selection of enterprise's large projects. Liao et al. [18] maked a survey on HFLTSs and reviewed the decision making process with hesitant fuzzy linguistic preference (HFLP) relations. Liao et al. [19] proposed the hesitant fuzzy linguistic preference utility set (HFLPUS) and applied the HFLP utility TOPSIS approach to choose the best fire rescue alternative. One thing that they have in common is that they use the subscript of linguistic terms directly in the process of operations, which may cause a loss of information. In order to overcome this problem, the linguistic scale function was introduced by Wang et al. [23], which can assign different numerical values to the linguistic terms set under different circumstances. The linguistic scale function can reflect the preferences of the decision makers in different environments. Since it was put forward, many scholars have studied this subject. For example, Wang et al. [24] presented the Hausdorff distance between the hesitant fuzzy linguistic numbers (HFLNs), based on the linguistic scale function, and developed the TOPSIS and TODIM approaches to it. Liu et al. [25] proposed a distance measure of HFLTSs, which also included the linguistic scale function. Furthermore, Liu et al. [26] proposed the intuitionistic fuzzy linguistic cosine similarity measure and the interval-valued intuitionistic fuzzy linguistic cosine similarity measure, they all contain the linguistic scale function. The research on this field has developed rapidly.

From another perspective, the similarity measure is also an important aspect in MCDM problems, which can measure the similarity degree between two different alternatives. It has been widely studied in the past few years. For example, Song et al. [27] considered the similarity measure between IFSs, and proposed the corresponding distance measure between IF belief functions. Liao et al. [9] presented some similarity measures and distance measures between HFLTSs; Lee et al. [10] proposed a similarity measure based on likelihood relations. For the other studies about the similarity measure, we can refer to [11–13]. The cosine similarity measure is also a significant similarity measure; it can be expressed as the inner product of two vectors divided by the product of their lengths [28]. Some scholars have studied the cosine similarity measure [29–31]. For instance, Ye [29] introduced a weighted cosine similarity measure between IFSs and they applied it to rank the alternative. Furthermore, Ye [30] presented the cosine similarity measure between interval-valued fuzzy sets (IVFSs) with risk preference, and altered its decision making method depending on decision makers' preferences. Liao et al. [31] defined the cosine similarity measure between HFLTSs and extended the TOPSIS approach and VIKOR approach to the cosine distance measure. It is already known that the cosine similarity measure proposed by Liao et al. [31] is not a regular similarity measure (because it is not satisfied with the axiom of the similarity measure; the example can be seen in Section 3). If it is applied in MCDM problems, it may cause the decision information to be distorted. Furthermore, the cosine similarity measure defined by Liao et al. [31] used the subscript of linguistic terms directly in process of operations; they did not consider the semantic decision environment, which may cause a loss of information in the decision making process.

Therefore, this paper introduces a new method to construct a similarity measure between HFLTSs; the main motivations and contributions of the paper are given as follows:

(1) In order to overcome the disadvantage of the similarity measure proposed by Liao et al. [31], a new similarity measure combining the existing cosine similarity measure [31] and the Euclidean distance measure of HFLTSs is proposed in this paper, which can improve the accuracy of the calculation to some extent.


The reminder of the paper is given as follows: the background on the MCDM problems, some concepts of LTS and HFLTS, the existing similarity measures of HFLTSs, and the linguistic scale function are reviewed in Section 2. In Section 3, a new score function of HFLTS based on the linguistic scale function, and a new approach to construct the similarity measure of HFLTSs, are presented. The corresponding distance measure is also constructed based on the relationship between the distance measure and the regular similarity measure. In Section 4, we extend the TOPSIS method to the proposed distance measure. In Section 5, a numerical example is given to illustrate the feasibility of the proposed method, and the same numerical example is examined to compare with other methods. Some conclusions and future research are proposed in Section 6.
