Research Article  Open Access
Pairwise Comparison and Distance Measure of Hesitant Fuzzy Linguistic Term Sets
Abstract
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 decisionmaking 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 much more simple. Finally, the pairwise comparison and distance measure methods are utilized to develop two multicriteria decisionmaking approaches under hesitant fuzzy linguistic environments. The results analysis shows that our methods in this paper are more reasonable.
1. Introduction
Since Zadeh introduced fuzzy sets [1] in 1965, several extensions of this concept have been developed, such as type2 fuzzy sets [2, 3] and interval type2 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 [14–18] (Cloud models are similar to nonstationary fuzzy sets and type2 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 [21–23], 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 [25–28], data mining [29], clinical diagnosis [30, 31], and subjective evaluation [32–37], especially in decisionmaking [38–49]. Usually, linguistic terms (words) are represented by fuzzy sets [50], type2 fuzzy sets [51], interval type2 fuzzy sets [52–54], 2tuple 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 an expert hesitating among several values for a linguistic variable. For example, when people assess a qualitative criterion, they prefer to use a linguistic one such as “between medium and very high” which contains several linguistic terms , rather than a single linguistic term. In practical decisionmaking process, uncertainty and hesitancy are usually unavoidable problems. The HFLTSs can deal with such uncertainty and hesitancy more objectively, and thus it is very necessary to develop some theories about HFLTSs.
Comparisons and distance measures used for measuring the deviations of different arguments are fundamentally important in a variety of applications. In the existing literature, there are a number of studies on distance measures for intuitionistic fuzzy sets [57–60], intervalvalued intuitionistic fuzzy sets [61], hesitant fuzzy sets [62, 63], linguistic values [64, 65], and so forth. Nevertheless, an HFLTS is a linguistic term subset, and the comparison among these elements is not simple. In [56], Rodriguez et al. introduced the concept of envelope for an HFLTS and then ranked HFLTSs using the preference degree method of interval values [66]. But, because an HFLTS is a set of discrete linguistic terms, it may seem problematical using the preference degree method for continuous interval to compare these discrete terms of HFLTSs. Up to now, just a few research has been done on the distance measure of HFLTSs [67]. Consequently, it is very necessary to develop some comparison methods and distance measure methods for HFLTSs. In [67], to calculate the distance of two HFLTSs, Liao et al. extend the shorter HFLTS by adding any value in it until it has the same length of the longer one according to the decisionmaker’s preferences and actual situations. In this paper, we present a new comparison method of 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 without adding any value. Finally, we utilize the comparison method and distance measure method to develop some approaches to solve the multicriteria decisionmaking problems under hesitant fuzzy linguistic environments.
The rest of the paper is organized as follows. In Section 2, the concepts of hesitant fuzzy sets and HFLTSs are introduced; also the defects of the previous comparison method for HFLTSs are analyzed according to an example. Section 3 describes the comparison and distance measure of HFLTSs based on the proposed pairwise comparison method. In Section 4, a multicriteria decisionmaking problem is shown to illustrate the detailed processes and effectiveness of two ranking methods which are based on the comparisons and distance measures of HFLTSs, respectively. Finally, Section 5 draws our conclusions and presents suggestions for future research.
2. 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 that when applied to returns a subset of .
To be easily understood, Zhu et al. [68] represented the HFS as the following mathematical symbol: where is a set of some values in , denoting the possible membership degrees of the element to the set . Liao et al. [67] called a hesitant fuzzy element (HFE).
Example 2. Let ; then is an HFE.
Definition 3 (see [69]). For an HFE , the score function of is defined as where is the number of the elements in .
For two HFEs and , if , then is superior to , denoted by ; if , then is indifferent with , denoted by .
Example 4. Assume that we have three HFEs, , , and ; then according to the score function of HFE, (2), and Definition 3, we have , , and . Thus, ; that is, the ranking is .
The concept of HFS is very useful to express people’s hesitancy in daily life. So, since it was introduced, more and more decisionmaking theories and methods under hesitant fuzzy environment have been developed [56, 62, 63, 67–73].
2.2. 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 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 .
It is required that the linguistic term set should satisfy the following characteristics:(1)the set is ordered: , if and only if ;(2)there is a negation operator: .
Example 6. Let be a linguistic term set, , , , , , , and two different HFLTSs might be and .
Definition 7. One defines the number of linguistic terms in the HFLTS as the cardinality of , denoted by . In Example 6, and .
Definition 8 (see [56]). The lower bound and upper bound of the HFLTS are defined as and .
Definition 9 (see [56]). The envelope of the HFLTS , , is defined as the linguistic interval ], where Ind provides the index of the linguistic term; that is, . In Example 6, and .
Based on the definition of envelope, Rodriguez et al. [56] compare two HFLTSs using the comparison method between two numerical intervals introduced by Wang et al. [66].
Definition 10 (see [66]). Letting and be two intervals, the preference degree of over (or ) is defined as and the preference degree of over (or ) is defined as
Example 11. Let , , and be three different HFLTSs on . According to Definition 9, we have , , and . The preference degrees calculated by Definition 10, (3), and (4) are
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.
The result indicates that is absolutely superior to . In fact, both and contain the linguistic term . It means that the value of a linguistic variable may be equal in these two cases. Thus, it is unreasonable to say that is absolutely superior to .
The result , meaning that when compared with , the two HFLTSs and are identical. In fact, is more superior to compared to to . 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.
3. 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 decisionmaking model, we can simply consider [49, 65]: .
Definition 12. Letting be two single linguistic terms, then we call the distance between and .
The distance measure between and has a definite physical implication and reflects the relative position and distance between and . If , then . If , then . If , then .
Theorem 13. Letting be three linguistic terms, then(1); (2); (3).
Proof. They are straightforward and thus omitted.
3.2. 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.
Definition 14. Letting and be two HFLTSs on , then one defines the pairwise comparison matrix between and as follows:
Remark 15. The number of linguistic terms in the two HFLTSs, and , may be unequal; that is, . To deal with such situations, usually it is necessary to extend the shorter one by adding the stated value several times in it [62, 63], while our pairwise comparison method does not require this step.
Remark 16. From Definition 14, we have , where is the transpose operator of matrix.
Example 17. Let and be two HFLTSs on . According to Definition 14, the comparison matrix between and is
Definition 18. Letting be the pairwise comparison matrix between and , the preference relations of and are defined as follows:
It is obvious that . We say that is superior to with the degree of , denoted by ; is equal to with the degree of , denoted by ; and is inferior to with the degree of , denoted by .
Considering Example 17, by Definition 18, (10), the preference relations of and were calculated as , , and . Thus, the comparison results are , , and .
3.3. Distance Measure of HFLTSs
Definition 19. Letting be the pairwise comparison matrix between and , the distance between and is defined as the average value of the pairwise comparison matrix:
Considering Example 17, one has .
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 decisionmaker 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 , and 3.5.
Theorem 22. Letting be an HFLTS on , then
Proof. It is straightforward and thus omitted.
Theorem 23. Letting and be two HFLTSs on , the distance between and defined by the average value of their pairwise comparison matrix is equal to the distance of the two average values of and ; that is, the distance between and can be easily obtained by
Proof. From Definitions 19 and 14, we have which completes the proof of Theorem 23.
Considering Example 17, we have .
By Theorem 23, we can easily obtain the following corollary.
Corollary 24. Letting , and be three HFLTSs on , then(1); (2); (3).
Proof. They are straightforward and thus omitted.
If (or ), then we say that is superior to with the distance of , denoted by ; if (or ), then we say that is indifferent to , denoted by ; if (or ), then we say that is inferior to with the distance of , denoted by .
4. Multicriteria DecisionMaking 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 decisionmaking 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.
Example 25 ([see [56]). Let be a set of alternatives, a set of criteria defined for each alternative, and , , , , , the linguistic term set that is used to generate the linguistic expressions. The assessments that are provided in such a problem are shown in Table 1 and they are transformed into HFLTSs as shown in Table 2.

4.1. Multicriteria DecisionMaking Based on the Comparisons of HFLTSs
Step 1. Considering each criterion , calculate the preference degrees between all the alternatives .
Considering criterion , , and , so the preference degrees about criterion calculated using the comparison method of HFLTSs as described in Section 3.2 are , , ; , , ; , , .
Considering criterion , , , and , so the preference degrees about criterion calculated using the comparison method of HFLTSs as described in Section 3.2 are , , ; , , ; , , .
Considering criterion , , , and , so the preference degrees about criterion calculated using the comparison method of HFLTSs as described in Section 3.2 are , , ; , , ; , , .
Step 2. Aggregate the preference relations using the weighted average method: , , and , where is the weight of criterion , and . In this paper, , . Thus, the final preference relations are , , ; , , ; , , .
Step 3. Rank the alternatives using the nondominance choice degree method as described in [56]. From the results of Step 2, it can be easily obtained that
Thus, , , and . Finally, the ranking of alternatives is .
4.2. Multicriteria DecisionMaking Based on the Distance Measures of HFLTSs
Step 1. Considering each criterion , calculate the average values of HFLTSs for all the alternatives . The results are shown in Table 3.

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 .
4.3. Results Analysis
In [56], the ranking of alternatives is , while both methods in this paper are . Note that the practical decisionmaking problem is quite different from other applications where wellestablished measures can be used to quantify the performance for validation. In decisionmaking, 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 and . Considering criterion , the original assessments of and are “between and ” and “greater than h,” respectively, so it is obviously about criterion . Considering criterion , the original assessments of and are “between and vh” and “between vl and l,” respectively, so this time . Considering criterion , the original assessments of and are “” and “greater than ,” respectively, so again. Summarily, occurs twice, while only once. Thus, we believe that our result is more plausible.
5. Conclusion
The comparison and distance measure of HFLTSs are fundamentally important in many decisionmaking 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 decisionmaking 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 decisionmaking problems will be explored. We will also investigate how to obtain the weights of criteria under hesitant fuzzy linguistic environments.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
References
 L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965. View at: Google Scholar  MathSciNet
 M. Mizumoto and K. Tanaka, “Some properties of fuzzy sets of type $2$,” Information and Control, vol. 31, no. 4, pp. 312–340, 1976. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 J. M. Mendel and R. I. B. John, “Type2 fuzzy sets made simple,” IEEE Transactions on Fuzzy Systems, vol. 10, no. 2, pp. 117–127, 2002. View at: Publisher Site  Google Scholar
 J. M. Mendel and H. Wu, “Type2 fuzzistics for symmetric interval type2 fuzzy sets: part 1, forward problems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 6, pp. 781–792, 2006. View at: Publisher Site  Google Scholar
 D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, NY, USA, 1980. View at: MathSciNet
 K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986. View at: Publisher Site  Google Scholar  MathSciNet
 K. T. Atanassov, “More on intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 33, no. 1, pp. 37–45, 1989. View at: Publisher Site  Google Scholar  MathSciNet
 K. T. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 31, no. 3, pp. 343–349, 1989. View at: Publisher Site  Google Scholar  MathSciNet
 W. Gau and D. J. Buehrer, “Vague sets,” IEEE Transactions on Systems, Man and Cybernetics, vol. 23, no. 2, pp. 610–614, 1993. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. Bustince and P. Burillo, “Vague sets are intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 79, no. 3, pp. 403–405, 1996. View at: Publisher Site  Google Scholar  MathSciNet
 R. R. Yager, “On the theory of bags,” International Journal of General Systems, vol. 13, no. 1, pp. 23–37, 1986. View at: Publisher Site  Google Scholar  MathSciNet
 S. Miyamoto, “Remarks on basics of fuzzy sets and fuzzy multisets,” Fuzzy Sets and Systems, vol. 156, no. 3, pp. 427–431, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 J. M. Garibaldi, M. Jaroszewski, and S. Musikasuwan, “Nonstationary fuzzy sets,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 4, pp. 1072–1086, 2008. View at: Publisher Site  Google Scholar
 D. Li, J. Han, X. Shi, and M. C. Chan, “Knowledge representation and discovery based on linguistic atoms,” KnowledgeBased Systems, vol. 10, no. 7, pp. 431–440, 1998. View at: Publisher Site  Google Scholar
 D. Li, C. Liu, and W. Gan, “A new cognitive model: cloud model,” International Journal of Intelligent Systems, vol. 24, no. 3, pp. 357–375, 2009. View at: Publisher Site  Google Scholar
 X. Yang, L. Zeng, F. Luo, and S. Wang, “Cloud hierarchical analysis,” Journal of Information & Computational Science, vol. 7, no. 12, pp. 2468–2477, 2010. View at: Google Scholar
 X. Yang, L. Zeng, and R. Zhang, “Cloud Delphi method,” International Journal of Uncertainty, Fuzziness and KnowlegeBased Systems, vol. 20, no. 1, pp. 77–97, 2012. View at: Publisher Site  Google Scholar
 X. Yang, L. Yan, and L. Zeng, “How to handle uncertainties in AHP: the Cloud Delphi hierarchical analysis,” Information Sciences, vol. 222, pp. 384–404, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 V. Torra, “Hesitant fuzzy sets,” International Journal of Intelligent Systems, vol. 25, no. 6, pp. 529–539, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 V. Torra and Y. Narukawa, “On hesitant fuzzy sets and decision,” in Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 1378–1382, Jeju Island, Korea, August 2009. View at: Publisher Site  Google Scholar
 L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—I,” Information Sciences, vol. 3, pp. 199–249, 1975. View at: Google Scholar  MathSciNet
 L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—II,” Information Sciences, vol. 4, pp. 301–357, 1975. View at: Google Scholar  MathSciNet
 L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning—III,” Information Sciences, vol. 9, no. 1, pp. 43–80, 1975. View at: Google Scholar  MathSciNet
 J. Ma, D. Ruan, Y. Xu, and G. Zhang, “A fuzzyset approach to treat determinacy and consistency of linguistic terms in multicriteria decision making,” International Journal of Approximate Reasoning, vol. 44, no. 2, pp. 165–181, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 G. Bordogna and G. Pasi, “A fuzzy linguistic approach generalizing boolean information retrieval: a model and its evaluation,” Journal of the American Society for Information Science, vol. 44, no. 2, pp. 70–82, 1993. View at: Google Scholar
 G. Bordogna and G. Pasi, “An ordinal information retrieval model,” International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, vol. 9, supplement 1, 2001. View at: Publisher Site  Google Scholar  MathSciNet
 J. Kacprzyk and S. Zadrozny, “Linguistic database summaries and their protoforms: towards natural language based knowledge discovery tools,” Information Sciences, vol. 173, no. 4, pp. 281–304, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 E. HerreraViedma and A. G. LópezHerrera, “A model of an information retrieval system with unbalanced fuzzy linguistic information,” International Journal of Intelligent Systems, vol. 22, no. 11, pp. 1197–1214, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. Ishibuchi, T. Nakashima, and M. Nii, Classification and Modeling with Linguistic Information Granules: Advanced Approaches to Linguistic Data Mining, Springer, Berlin, Germany, 2004.
 R. Degani and G. Bortolan, “The problem of linguistic approximation in clinical decision making,” International Journal of Approximate Reasoning, vol. 2, no. 2, pp. 143–162, 1988. View at: Publisher Site  Google Scholar
 E. Sanchez, “Truthqualification and fuzzy relations in natural languages, application to medical diagnosis,” Fuzzy Sets and Systems, vol. 84, no. 2, pp. 155–167, 1996. View at: Publisher Site  Google Scholar  MathSciNet
 H. Lee, “Applying fuzzy set theory to evaluate the rate of aggregative risk in software development,” Fuzzy Sets and Systems, vol. 79, no. 3, pp. 323–336, 1996. View at: Publisher Site  Google Scholar
 L. Martínez, “Sensory evaluation based on linguistic decision analysis,” International Journal of Approximate Reasoning, vol. 44, no. 2, pp. 148–164, 2007. View at: Publisher Site  Google Scholar
 L. Martínez, J. Liu, D. Ruan, and J. Yang, “Dealing with heterogeneous information in engineering evaluation processes,” Information Sciences, vol. 177, no. 7, pp. 1533–1542, 2007. View at: Publisher Site  Google Scholar
 J. Lu, Y. Zhu, X. Zeng, L. Koehl, J. Ma, and G. Zhang, “A linguistic multicriteria group decision support system for fabric hand evaluation,” Fuzzy Optimization and Decision Making, vol. 8, no. 4, pp. 395–413, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 R. de Andrés, M. Espinilla, and L. Martínez, “An extended hierarchical linguistic model for managing integral evaluation,” International Journal of Computational Intelligence Systems, vol. 3, no. 4, pp. 486–500, 2010. View at: Publisher Site  Google Scholar
 D. Wu and J. M. Mendel, “Computing with words for hierarchical decision making applied to evaluating a weapon system,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 3, pp. 441–460, 2010. View at: Publisher Site  Google Scholar
 F. Herrera and E. HerreraViedma, “Choice functions and mechanisms for linguistic preference relations,” European Journal of Operational Research, vol. 120, no. 1, pp. 144–161, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. Herrera, E. HerreraViedma, and L. Martínez, “A fusion approach for managing multigranularity linguistic term sets in decision making,” Fuzzy Sets and Systems, vol. 114, no. 1, pp. 43–58, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 F. Herrera and L. Martínez, “A 2tuple fuzzy linguistic representation model for computing with words,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 6, pp. 746–752, 2000. View at: Google Scholar
 B. Arfi, “Fuzzy decision making in politics: a linguistic fuzzyset approach,” Political Analysis, vol. 13, no. 1, pp. 23–56, 2005. View at: Publisher Site  Google Scholar
 V. Huynh and Y. Nakamori, “A satisfactoryoriented approach to multiexpert decisionmaking with linguistic assessments,” IEEE Transactions on Systems, Man, and Cybernetics B: Cybernetics, vol. 35, no. 2, pp. 184–196, 2005. View at: Publisher Site  Google Scholar
 D. BenArieh and Z. Chen, “Linguistic group decisionmaking: opinion aggregation and measures of consensus,” Fuzzy Optimization and Decision Making, vol. 5, no. 4, pp. 371–386, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 J. L. GarcíaLapresta, “A general class of simple majority decision rules based on linguistic opinions,” Information Sciences, vol. 176, no. 4, pp. 352–365, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 J. Wang and J. Hao, “Fuzzy linguistic PERT,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 2, pp. 133–144, 2007. View at: Publisher Site  Google Scholar
 T. C. Wang and Y. H. Chen, “Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP,” Information Sciences, vol. 178, no. 19, pp. 3755–3765, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Dong, Y. Xu, and S. Yu, “Linguistic multiperson decision making based on the use of multiple preference relations,” Fuzzy Sets and Systems, vol. 160, no. 5, pp. 603–623, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 P. D. Liu, “A novel method for hybrid multiple attribute decision making,” KnowledgeBased Systems, vol. 22, no. 5, pp. 388–391, 2009. View at: Publisher Site  Google Scholar
 Y. Dong, W. Hong, Y. Xu, and S. Yu, “Selecting the individual numerical scale and prioritization method in the analytic hierarchy process: a 2Tuple fuzzy linguistic approach,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 1, pp. 13–25, 2011. View at: Publisher Site  Google Scholar
 L. A. Zadeh, “Fuzzy logic = computing with words,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 2, pp. 103–111, 1996. View at: Publisher Site  Google Scholar
 J. M. Mendel, “Fuzzy sets for words: a new beginning,” in Proceedings of the IEEE International conference on Fuzzy Systems, pp. 37–42, St. Louis, Mo, USA, May 2003. View at: Google Scholar
 J. M. Mendel, “Computing with words and its relationships with fuzzistics,” Information Sciences, vol. 177, no. 4, pp. 988–1006, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 F. Liu and J. M. Mendel, “Encoding words into interval type2 fuzzy sets using an interval approach,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 6, pp. 1503–1521, 2008. View at: Publisher Site  Google Scholar
 S. Coupland, J. M. Mendel, and D. Wu, “Enhanced interval approach for encoding words into interval type2 fuzzy sets and convergence of the word FOUs,” in Proceedings of the 6th IEEE World Congress on Computational Intelligence (WCCI '10), pp. 1–8, Barcelona, Spain, July 2010. View at: Publisher Site  Google Scholar
 L. Martínez and F. Herrera, “An overview on the 2tuple linguistic model for computing with words in decision making: extensions, applications and challenges,” Information Sciences, vol. 207, pp. 1–18, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 R. M. Rodriguez, L. Martinez, and F. Herrera, “Hesitant fuzzy linguistic term sets for decision making,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 1, pp. 109–119, 2012. View at: Publisher Site  Google Scholar
 E. Szmidt and J. Kacprzyk, “Distances between intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 114, no. 3, pp. 505–518, 2000. View at: Publisher Site  Google Scholar  MathSciNet
 P. Grzegorzewski, “Distances between intuitionistic fuzzy sets and/or intervalvalued fuzzy sets based on the Hausdorff metric,” Fuzzy Sets and Systems, vol. 148, no. 2, pp. 319–328, 2004. View at: Publisher Site  Google Scholar  MathSciNet
 W. Wang and X. Xin, “Distance measure between intuitionistic fuzzy sets,” Pattern Recognition Letters, vol. 26, no. 13, pp. 2063–2069, 2005. View at: Publisher Site  Google Scholar
 Z. S. Xu and J. Chen, “An overview of distance and similarity measures of intuitionistic fuzzy sets,” International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, vol. 16, no. 4, pp. 529–555, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. S. Xu, “A method based on distance measure for intervalvalued intuitionistic fuzzy group decision making,” Information Sciences, vol. 180, no. 1, pp. 181–190, 2010. View at: Publisher Site  Google Scholar
 Z. Xu and M. Xia, “Distance and similarity measures for hesitant fuzzy sets,” Information Sciences, vol. 181, no. 11, pp. 2128–2138, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 Z. Xu and M. Xia, “On distance and correlation measures of hesitant fuzzy information,” International Journal of Intelligent Systems, vol. 26, no. 5, pp. 410–425, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Z. S. Xu, “Deviation measures of linguistic preference relations in group decision making,” Omega, vol. 33, no. 3, pp. 249–254, 2005. View at: Publisher Site  Google Scholar
 Y. Xu and H. Wang, “Distance measure for linguistic decision making,” Systems Engineering Procedia, vol. 1, pp. 450–456, 2011. View at: Google Scholar
 Y. M. Wang, J. B. Yang, and D. L. Xu, “A preference aggregation method through the estimation of utility intervals,” Computers and Operations Research, vol. 32, no. 8, pp. 2027–2049, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. Liao, Z. Xu, and X. Zeng, “Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multicriteria decision making,” Information Sciences, vol. 271, pp. 125–142, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 B. Zhu, Z. Xu, and M. Xia, “Hesitant fuzzy geometric Bonferroni means,” Information Sciences, vol. 205, pp. 72–85, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 M. Xia and Z. Xu, “Hesitant fuzzy information aggregation in decision making,” International Journal of Approximate Reasoning, vol. 52, no. 3, pp. 395–407, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. Gu, Y. Wang, and B. Yang, “A method for hesitant fuzzy multiple attribute decision making and its application to risk investment,” Journal of Convergence Information Technology, vol. 6, no. 6, pp. 282–287, 2011. View at: Publisher Site  Google Scholar
 M. Xia, Z. Xu, and N. Chen, “Some hesitant fuzzy aggregation operators with their application in group decision making,” Group Decision and Negotiation, vol. 22, no. 2, pp. 259–279, 2013. View at: Publisher Site  Google Scholar
 G. Wei, “Hesitant fuzzy prioritized operators and their application to multiple attribute decision making,” KnowledgeBased Systems, vol. 31, pp. 176–182, 2012. View at: Publisher Site  Google Scholar
 B. Zhu, Z. Xu, and M. Xia, “Dual hesitant fuzzy sets,” Journal of Applied Mathematics, vol. 2012, Article ID 879629, 13 pages, 2012. View at: Publisher Site  Google Scholar  MathSciNet
Copyright
Copyright © 2014 HanChen Huang and Xiaojun Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.