Pairwise Comparison and Distance Measure of Hesitant Fuzzy Linguistic Term Sets

A hesitant fuzzy linguistic term set (HFLTS), allowing experts using several possible linguistic terms to assess a qualitative linguistic variable, is very useful to express people’s hesitancy in practical decision-making problems. Up to now, a little research has been done on the comparison and distance measure of HFLTSs. In this paper, we present a comparison method for HFLTSs based on pairwise comparisons of each linguistic term in the two HFLTSs. Then, a distance measure method based on the pairwise comparison matrix of HFLTSs is proposed, and we prove that this distance is equal to the distance of the average values of HFLTSs, which makes the distance measure muchmore simple. Finally, the pairwise comparison and distance measure methods are utilized to develop two multicriteria decision-making approaches under hesitant fuzzy linguistic environments. The results analysis shows that our methods in this paper are more reasonable.


Introduction
Since Zadeh introduced fuzzy sets [1] in 1965, several extensions of this concept have been developed, such as type-2 fuzzy sets [2,3] and interval type-2 fuzzy sets [4], type- fuzzy sets [5], intuitionistic fuzzy sets [6,7] and intervalvalued intuitionistic fuzzy sets [8], vague sets [9] (vague sets are intuitionistic fuzzy sets [10]), fuzzy multisets [11,12], nonstationary fuzzy sets [13], Cloud models [14][15][16][17][18] (Cloud models are similar to nonstationary fuzzy sets and type-2 fuzzy sets), and hesitant fuzzy sets [19,20].In the real world, there are many situations in which problems must deal with qualitative aspects represented by vague and imprecise information.So, in these situations, often the experts are more accustomed to express their assessments using linguistic terms rather than numerical values.In [21][22][23], Zadeh introduced the concept of linguistic variable as "a variable whose values are not numbers but words or sentences in a natural or artificial language." Linguistic variable provides a means of approximate characterization of phenomena which are too complex or too ill defined to be amenable to description in conventional quantitative ways.Since then, fuzzy sets and linguistic variables have been widely used in describing linguistic information as they can efficiently represent people's qualitative cognition of an object or a concept [24].Thus, linguistic approaches have been so far used successfully in a wide range of applications, such as information retrieval [25][26][27][28], data mining [29], clinical diagnosis [30,31], and subjective evaluation [32][33][34][35][36][37], especially in decision-making [38][39][40][41][42][43][44][45][46][47][48][49].Usually, linguistic terms (words) are represented by fuzzy sets [50], type-2 fuzzy sets [51], interval type-2 fuzzy sets [52][53][54], 2-tuple linguistic model [40,55], and so forth.In these linguistic models, an expert generally provides a single linguistic term as an expression of his/her knowledge.However, just as Rodriguez et al. [56] pointed out, the expert may think of several terms at the same time or look for a more complex linguistic term that is not defined in the linguistic term set to express his/her opinion.In order to cope with this situation, they recently introduced the concept of hesitant fuzzy linguistic term sets (HFLTSs) [56] under the idea of hesitant fuzzy sets introduced in [19,20].
Similarly to a hesitant fuzzy set which permits the membership having a set of possible values, an HFLTS allows

Preliminaries
2.1.Hesitant Fuzzy Sets.Hesitant fuzzy sets (HFSs) were first introduced by Torra [19] and Torra and Narukawa [20].The motivation is that when determining the membership degree of an element into a set, the difficulty is not because we have a margin of error (such as an interval) but because we have several possible values.
Definition 1 (see [19]).Let  be a fixed set; a hesitant fuzzy set (HFS) on  is in terms of a function ℎ that when applied to  returns a subset of [0, 1].
To be easily understood, Zhu et al. [68] represented the HFS as the following mathematical symbol: where ℎ() is a set of some values in [0, 1], denoting the possible membership degrees of the element  ∈  to the set . Liao et al. [67] called ℎ() a hesitant fuzzy element (HFE).
Definition 3 (see [69]).For an HFE ℎ, the score function of ℎ is defined as where #ℎ is the number of the elements in ℎ.

Hesitant Fuzzy Linguistic Term Sets.
Similarly to the HFS, an expert may hesitate among several linguistic terms, such as "between medium and very high" or "lower than medium, " to assess a qualitative linguistic variable.To deal with such situations, Rodriguez et al. [56] introduced the concept of hesitant fuzzy linguistic term sets (HFLTSs).
Definition 5 (see [56]).Suppose that  = { 0 , . . .,   } is a finite and totally ordered discrete linguistic term set, where   represents a possible value for a linguistic variable.An HFLTS,   , is defined as an ordered finite subset of the consecutive linguistic terms of .
Definition 10 (see [66]).Letting  = [ 1 ,  2 ] and  = [ 1 ,  2 ] be two intervals, the preference degree of  over  (or  > ) is defined as and the preference degree of  over  (or  > ) is defined as From Example 11 mentioned above, it can be observed that when we compare two HFLTSs using the preference degree method, there exist two defects as follows.
(1) The result ( to  1  .Thus, using the preference degree method to compare HFLTSs may result in losing some important information.
Based on the analysis mentioned above, we think that it is not suitable to compare discrete linguistic terms in HFLTSs using the comparison method for continuous numerical intervals.By the definition of an HFLTS, we know that every linguistic term in it is a possible value of the linguistic information.And noting that, the two HFLTSs for comparing may have different lengths.So, when comparing two HFLTSs, it needs pairwise comparisons of each linguistic term in them.

Comparison and Distance
Measure of HFLTSs 3.1.Distance between Two Single Linguistic Terms.Let   ,   ∈  be two linguistic terms.Xu [64] defined the deviation measure between   and   as follows: where  is the cardinality of ; that is,  = ||.
If only one preestablished linguistic term set  is used in a decision-making model, we can simply consider [49,65]: Definition 12. Letting   ,   ∈  be two single linguistic terms, then we call  (  ,   ) =   −   =  −  (7) the distance between   and   .

Comparison of HFLTSs.
The comparison of HFLTSs is necessary in many problems, such as ranking and selection.However, an HFLTS is a linguistic term subset which contains several linguistic terms, and the comparison among HFLTSs is not simple.Here, a new comparison method of HFLTSs, which is based on pairwise comparisons of each linguistic term in the two HFLTSs, is put forward.
To preserve all the given information, the discrete linguistic term set  is extended to a continuous term set  = {  |  ∈ [−, ]}, where  is a sufficiently large positive number.If   ∈ , then we call   an original linguistic term; otherwise, we call   a virtual linguistic term.
Remark 20.In general, the decision-maker uses the original linguistic terms to express his/her qualitative opinions, and the virtual linguistic terms can only appear in operations.
Definition 21.The average value of an HFLTS   is defined as This definition is similar to the score function of an HFE, Definition 3.
Considering Example 17, we have Aver( Proof.From Definitions 19 and 14, we have which completes the proof of Theorem 23.
Considering Example 17, we have (

Multicriteria Decision-Making Models Based on Comparisons and Distance Measures of HFLTSs
In this section, two new methods are presented for ranking and choice from a set of alternatives in the framework of multicriteria decision-making using linguistic information.
One is based on the comparisons and preference relations of HFLTSs and the other is based on the distance measure of HFLTSs.We adopt Example 5 in [56] (Example 25 in our paper) to illustrate the detailed processes of the two methods.
Step 2. Aggregate the average values using the weighted average method.The results are shown in Table 4.
Step 3. Rank the alternatives using the distance measure method.Thus, the ranking of alternatives is

Results Analysis.
In [56], the ranking of alternatives is  1 ≻  3 ≻  2 , while both methods in this paper are  3 ≻  1 ≻  2 .Note that the practical decision-making problem is quite different from other applications where wellestablished measures can be used to quantify the performance for validation.In decision-making, usually there is no ground truth data or quantitative measures to assess the performance of a method [37].This is why "plausibility" is used rather than "validation." Here, we analyze the original assessments about each criterion of alternatives  1 and  3 .Considering criterion  1 , the original assessments of  1 and  3 are "between vl and m" and "greater than h, " respectively, so it is obviously  3 ≻  1 about criterion  1 .Considering criterion  2 , the original assessments of  1 and  3 are "between h and vh" and "between vl and l, " respectively, so this time  1 ≻  3 .Considering criterion  3 , the original assessments of  1 and  3 are "h" and "greater than h, " respectively, so  3 ≻  1 again.Summarily,  3 ≻  1 occurs twice, while  1 ≻  3 only once.Thus, we believe that our result is more plausible.

Conclusion
The comparison and distance measure of HFLTSs are fundamentally important in many decision-making problems under hesitant fuzzy linguistic environments.From an example, we found that there existed two defects when comparing HFLTSs using the previous preference degree method.By analyzing the definition of an HFLTS, a new comparison method based on pairwise comparisons of each linguistic term in the two HFLTSs has been put forward.This comparison method does not need the assumption that the values in all HFLTSs are arranged in an increasing order and two HFLTSs have the same length when comparing them.Then, we have defined a distance measure method between HFLTSs based on pairwise comparisons.Further, we have proved that this distance is equal to the distance of the average values of HFLTSs, which makes the distance measure much simpler.Finally, two new methods for multicriteria decision-making in which experts provide their assessments by HFLTSs have been proposed.The encouraging results demonstrate that our methods in this paper are more reasonable.
In the future, the application of HFLTSs to group decision-making problems will be explored.We will also investigate how to obtain the weights of criteria under hesitant fuzzy linguistic environments.

Table 1 :
Assessments that are provided for the decision problem.

Table 2 :
Assessments transformed into HFLTSs. = { 1 ,  2 }, so the preference degrees about criterion  2 calculated using the comparison method of HFLTSs as described in Section 3.2 are  2 1 ,  2 }, and

Table 3 :
Average values of the assessments.

Table 4 :
Aggregation results of each alternative.