Information Sciences 271 (2014) 125–142

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Information Sciences

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Distance and similarity measures for hesitant fuzzy linguisticterm sets and their application in multi-criteria decision making

http://dx.doi.org/10.1016/j.ins.2014.02.1250020-0255/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Tel.: +86 25 84483382.E-mail addresses: liaohuchang@163.com (H. Liao), xuzeshui@263.net (Z. Xu), x.zeng@manchester.ac.uk (X.-J. Zeng).

Huchang Liao a,c, Zeshui Xu b,⇑, Xiao-Jun Zeng c

a Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, Chinab Business School, Sichuan University, Chengdu, Sichuan 610065, Chinac School of Computer Science, University of Manchester, Manchester M13 9PL, United Kingdom

a r t i c l e i n f o

Article history:Received 4 December 2012Received in revised form 20 February 2014Accepted 21 February 2014Available online 1 March 2014

Keywords:Hesitant fuzzy linguistic term setDistance measureSimilarity measureMulti-criteria decision making

a b s t r a c t

The hesitant fuzzy linguistic term sets (HFLTSs), which can be used to represent an expert’shesitant preferences when assessing a linguistic variable, increase the flexibility of elicitingand representing linguistic information. The HFLTSs have attracted a lot of attentionrecently due to their distinguished power and efficiency in representing uncertainty andvagueness within the process of decision making. To enhance and extend the applicabilityof HFLTSs, this paper investigates and develops different types of distance and similaritymeasures for HFLTSs. The paper first proposes a family of distance and similarity measuresbetween two HFLTSs. Then a variety of weighted or ordered weighted distance and similar-ity measures between two collections of HFLTSs are proposed and analyzed for discrete andcontinuous cases respectively. After that, the application of these measures to multi-crite-ria decision making problems is given. Based on the proposed distance and similarity mea-sures, the satisfaction degrees for different alternatives are established and are then used torank alternatives in multi-criteria decision making. Finally a practical example concerningthe evaluation of the quality of movies is given to illustrate the applicability and advantageof the proposed approach and the differences between the proposed distance and similar-ity measures.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Hesitant fuzzy sets (HFSs), which were first introduced by Torra [30] as an extended form of fuzzy sets, have attracted alot of attention recently due to their effectiveness and efficiency in representing uncertainty and vagueness[13–16,18,30,46,52]. The motivation for introducing HFSs was that it is sometimes difficult to determine the membershipdegree of an element to a set, and in some circumstances this difficulty is caused by a doubt between a few different values[30]. Since the HFS permits the membership degree of an element to a given set represented by several possible valuesbetween 0 and 1, it can express a decision maker’s hesitancy efficiently, especially when two or more sources of vaguenessappear simultaneously.

It should be noted that the HFS was introduced to handle the problems that are represented in quantitative situations. Inmany cases, however, uncertainty is produced by the vagueness of meanings whose nature is qualitative rather than quan-titative [4,8,9,24]. For example, when evaluating the ‘‘speed’’ of a car, the linguistic terms such as ‘‘fast’’, ‘‘very fast’’, ‘‘slow’’may be used; when evaluating the ‘‘performance’’ of a company, the terms such as ‘‘good’’, ‘‘medium’’, and ‘‘bad’’ can be used.

126 H. Liao et al. / Information Sciences 271 (2014) 125–142

For such cases, Zadeh [51] proposed the fuzzy linguistic approach, which has been extended into several different models,such as the linguistic model based on type-2 fuzzy sets [31], 2-tuple fuzzy linguistic representation model [10,20,21], theproportional 2-tuple model [32], and so on [6]. However, all these extended models have some serious limitation due tothe fact that they assess a linguistic variable by using a single linguistic term rather than following the information providedby decision makers regarding the linguistic variable. Because decision makers may consider several terms at the same timeor need a complex linguistic term, such a single linguistic term is often insufficient or very hard to be determined. Motivatedby HFSs and linguistic fuzzy sets, Rodríguez et al. [26] proposed the concept of the hesitant fuzzy linguistic term set (HFLTS),which provides a different and more powerful form to represent decision makers’ preferences in the decision making pro-cess. The HFLTS increases the flexibility and capability of eliciting and representing linguistic information. It permits decisionmakers to use several linguistic terms to assess a linguistic variable. Thus, it provides many advantages in depicting decisionmakers’ cognitions and preferences.

Rodríguez et al. [26] applied HFLTSs to multi-criteria linguistic decision making problems in which decision makers canprovide their assessments by linguistic expressions based on comparative terms, such as ‘‘between very low and medium’’, orby simple terms, such as ‘‘very low; low; medium; high; very high’’. By using HFLTSs and context-free grammar, Rodríguezet al. [27] then proposed a new linguistic group decision making model that facilitates the elicitation of flexible linguisticexpressions close to human being’s cognitive models for expressing linguistic preferences. Liu and Rodríguez [19] presenteda new representation of HFLTSs by means of a fuzzy envelope to carry out the computing with words process. Later, Zhu andXu [53] introduced the hesitant fuzzy linguistic preference relation (HFLPR) as a tool to collect and represent decision mak-ers’ preferences and then investigated the consistency of the HFLPR. In order to apply HFLTSs to solve multi-criteria decisionmaking problems more effectively, we shall pay more attention to the basic characteristics of HFLTSs, in particular distanceand similarity measures which are fundamentally important in many scientific fields, such as decision making, pattern rec-ognition, and machine learning [42,43,45,46]. In addition, these measures are also the basis of some well-known methods,such as TOPSIS, VIKOR, ELECTRE. Hence, in this paper, we focus on investigating the distance and similarity measures forHFLTSs, and then apply them to multi-criteria decision making within the context of hesitant fuzzy linguistic circumstances.

Based on this focus, the rest of this paper is organized as follows: Section 2 presents the concepts of the linguistic termsets and hesitant fuzzy linguistic term sets. In Section 3, we first review some known distance and similarity measures andthen give the definitions of distance and similarity measures for HFLTSs, based on which several distance and similarity mea-sures for two HFLTSs are introduced. In Section 4, we focus on the distance and similarity measures for two collections ofHFLTSs, and establish a variety of weighted distance and similarity measures for discrete and continuous cases respectively.Section 5 gives the application of the proposed distance and similarity measures to multi-criteria decision making. The sat-isfaction degrees of different alternatives are defined in order to rank alternatives. A practical example concerning the eval-uation of the quality of movies is then given to illustrate the applicability and advantage of the proposed approach and thedifferent distance and similarity measures.

2. Linguistic term sets and hesitant fuzzy linguistic term sets

2.1. Linguistic term sets

In the process of decision making, decision makers may feel comfortable and straightforward to provide their knowledgeby using linguistic terms that are close to human being’s cognitive processes. To model and manage such knowledge withuncertainty, the fuzzy linguistic approach which uses fuzzy set theory to model the linguistic information was proposed byZadeh in [51]. The linguistic variable, defined as ‘‘a variable whose values are not numbers but words or sentences in a natural orartificial language’’, enhances the flexibility and applicability of the decision models and provides good application results inmany different fields [25].

Definition 1 [51]. A linguistic variable is characterized by a quintuple ðH; TðHÞ;U;G;MÞ, where H is the name of variable;TðHÞ (or simply TÞ denotes the term set of H, i.e., the set of its linguistic values, U is a universe of discourse; G is a syntacticrule (which usually takes the form of a grammar) for generating the terms in TðHÞ; and M is a semantic rule for associatingeach linguistic value X with its meaning, MðXÞ, which is a fuzzy subset of U.

The definition reveals that a linguistic variable is actually established by its linguistic descriptors and semantics. There aredifferent ways to choose the linguistic descriptors and to define their semantics [8,9,25,26,48]. The commonly usedapproaches for selecting the linguistic descriptors include the ordered structure approach and context-free grammar approach.The definitions of their semantics can be accomplished in three ways: (1) semantics based on an ordered structure of thelinguistic term set; (2) semantics based on membership functions and a semantic rule; and (3) mixed semantics. We pay ourattention herein to the ordered structure approach and the semantics based on the ordered structure of the linguistic term set

By means of supplying directly the term set, the ordered structure approach defines the linguistic term set via consideringall the terms that are distributed on a scale [8,9,48]. The well-known set of seven linguistic terms is given as:S ¼ fs0 ¼ none; s1 ¼ very low; s2 ¼ low; s3 ¼ medium; s4 ¼ high; s5 ¼ very high; s6 ¼ perfectg. Meanwhile, some other lin-guistic term sets, which are distributed based on different scales, have been developed as well. For example, Xu[35,37,41] introduced a subscript-symmetric linguistic evaluation scale, which can be defined as follows:

H. Liao et al. / Information Sciences 271 (2014) 125–142 127

S ¼ fsaja ¼ �s; . . . ;�1;0;1;1; . . . ; sg ð1Þ

where the mid linguistic label s0 represents an assessment of ‘‘indifference’’, and the rest of them are placed symmetricallyaround it. s�s and ss are the lower and upper bounds of linguistic labels where s is a positive integer. S satisfies the followingconditions:

(1) If a > b, then sa > sb.(2) The negation operator is defined as: neg ðsaÞ ¼ s�a, especially, neg ðs0Þ ¼ s0.

For example, when s ¼ 3; S can be taken as (see Fig. 1):

S ¼ fs�3 ¼ none; s�2 ¼ very low; s�1 ¼ low; s0 ¼ medium; s1 ¼ high; s2 ¼ very high; s3 ¼ perfectg

The semantics based on the ordered structure of the linguistic term set introduces the semantics over the linguistic term set.The users provide their assessments by using an ordered linguistic term set. A linguistic term set of seven subscript-symmet-ric terms with its syntax and fuzzy semantics representation is graphically shown in Fig. 2.

When non-probabilistic uncertainty arises, the use of linguistic information facilitates decision makers’ preference elic-itation. The decision makers can use the fuzzy linguistic approach to express their assessments over a linguistic variable. Forexample, considering a person’s age as a linguistic variable, the linguistic term set could be given asTðageÞ ¼ fyoung; not young; very young; old; not old; very old; middleaged; not middle� aged; etc:g. In such a case, thenumerical variable age, whose values are the numbers 0;1;2;3; . . . ;100, constitutes what may be called the base variablefor age. The linguistic value (such as young) can be interpreted as a label for a fuzzy restriction on the values of the base var-iable. Such fuzzy restriction is characterized by a compatibility function associated each age in the interval ½0;100� with a realnumber in the interval ½0;1�, which represents the compatibility of that age with the label young. For instance, the compat-ibility degrees of the numerical ages 20, 25 and 35 with the linguistic value young might be 1, 0.9 and 0.1, respectively. Thecompatibility function is in fact the membership function in Zadeh’s fuzzy set theory [51]. The triangular membership func-tion (see Figs. 2 and 3) and Gaussian functions [41] are the commonly used ones.

It is noted that there is some limitation in the fuzzy linguistic approach. When we use some computation methods [6,8–10] to calculate linguistic values, the final results usually do not exactly match any of the initial linguistic terms. Then, anapproximation process must be employed to translate the results into the initial expression domain, which consequentlyproduces the loss of information. In order to preserve all given information, Xu [37] extended the discrete linguistic termset S to the continuous linguistic term set S ¼ fsaja 2 ½�q; q�g, where qðq > sÞ is a sufficiently large positive integer. In gen-eral, the linguistic term saðsa 2 SÞ is given by the decision maker, while the extended linguistic term (also named virtual lin-guistic term) �sað�sa 2 SÞ only appears in computation.

For any two linguistic terms sa; sb 2 S and k; k1; k2 2 ½0;1�, the following operational laws were introduced by Xu [35]:

(1) sa � sb ¼ saþb;(2) ksa ¼ ska;(3) ðk1 þ k2Þsa ¼ k1sa � k2sa;(4) kðsa � sbÞ ¼ ksa � ksb.

1s− 0s2s−3s− 1s 2s 3s

none very low low medium high very high perfect

Fig. 1. Subscript-symmetric linguistic term set Sðs ¼ 3Þ.

0 0.17 0.33 0.5 0.67 0.83 1

3s

none− 2s

very low− 1s

low− 0s

medium1s

high2s

very high3s

perfect

Fig. 2. The set of seven subscript-symmetric terms with its semantics.

0 0.17 0.33 0.5 0.67 0.83 1

3s

none− 2s

very low− 1s

low− 0s

medium1s

high2s

very high3s

perfect

1.3s

0.551 0.881

1.6s−

0.068 0.399

Fig. 3. Semantics of virtual linguistic terms. Notes: The bounds of terms are easy to be calculated. Taking s1:3 as an example, 0:551 ¼ 0:5þ ð1:3� 1Þ � 0:17and 0:881 ¼ 0:83þ ð1:3� 1Þ � 0:17.

128 H. Liao et al. / Information Sciences 271 (2014) 125–142

It is noted that the operational laws (1) and (2) hold only from a theoretical point of view, and we usually do not use themin practice because it makes no sense if we add the linguistic term ‘‘young’’ to the linguistic term ‘‘very young’’. When com-puting linguistic variables in practical applications, the fusing process is always associated with a weighting vector [41], thatis, the fusion is in fact a convex combination. For this reason, we only need to use the operational laws (3) and (4). Since s�s

and ss are the lower and upper bounds of the linguistic labels determined by the decision maker, it is impossible that theinitial value of a linguistic variable given by the decision maker is greater than s�s or lower than ss. Hence, the final resultsare still within the interval ½s�s; ss�.

One question that has triggered off some discussions over the virtual linguistic term set concerns their corresponding fuz-zy semantics representation and linguistic syntax [25]. In fact, we still can construct the mapping between virtual linguisticterms and their corresponding semantics. For example, if we obtain two virtual linguistic terms s1:3 and s�1:6, then thesemantics of them can be represented by the compatibility functions with the triangular formsCs1:3 ¼ ð0:551;0:721;0:881Þ and Cs�1:6 ¼ ð0:068;0:238;0:399Þ, which are shown in Fig. 3.

It should be noted that the (virtual) linguistic term set model is different from the Likert scale. The Likert scale is a psy-chometric scale commonly involved in the research based on survey questionnaires. The range of Likert scale only capturesthe discrete intensity of respondents’ feelings for a given item. Thus, the Likert scale cannot use a function to represent theintensity, while the (virtual) linguistic term set is a fuzzy linguistic approach, which utilizes the compatibility function toexpress the compatibility between the basic variable value and the linguistic term. For example, in Fig. 3, the compatibilityfunctions of the linguistic terms are ones with the triangular type Csa ¼ ðai; bi; ciÞ. Furthermore, the (virtual) linguistic termset is associated with a syntactic rule and a semantic rule, which makes it more complex than the Likert scale.

2.2. Hesitant fuzzy linguistic term sets

Hesitant fuzzy sets, which permit the membership degree of an element to a reference set represented by several possiblevalues, is a powerful structure in reflecting a decision maker’s hesitance.

Definition 2 [30]. Let X be a fixed set, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returnsa subset of ½0;1�.

To be easily understood, Xu and Xia [33,46] expressed the HFS by a mathematical symbol:

E ¼ fhx; hEðxÞijx 2 Xg ð2Þ

where hEðxÞ is a subset of ½0;1�, denoting the possible membership degrees of an element x 2 X to set E.Similar to the situations of HFSs where a decision maker may hesitate between several possible values as the membership

degree when evaluating an alternative, in a qualitative circumstance, a decision maker may hesitate between several termsto assess a linguistic variable. Hence, motivated by the idea of HFSs, Rodríguez et al. [26] introduced the hesitant fuzzylinguistic term set (HFLTS), whose envelope is an uncertain linguistic variable [34].

Definition 3 [26]. Let S ¼ fs0; . . . ; ssg be a linguistic term set, a hesitant fuzzy linguistic term set (HFLTS), HS, is an orderedfinite subset of the consecutive linguistic terms of S.

For the linguistic term set S ¼ fs0; . . . ; ssg in Definition 3, as its subscripts are not symmetric, then some problems willarise. For example, if we take S ¼ fs0 ¼ none; s1 ¼ very low; s2 ¼ low; s3 ¼ medium; s4 ¼ high; s5 ¼ very high; s6 ¼ perfectg,then s2 � s4 ¼ s6, the aggregated result of linguistic terms ‘‘low’’ and ‘‘high’’ is ‘‘perfect’’. This is not coincident with ourintuition. To circumvent this issue, we can replace the linguistic term set S ¼ fs0; . . . ; ssg by the subscript-symmetriclinguistic term set S ¼ fsaja ¼ �s; . . . ;�1;0;1; . . . ; sg. It is worth pointing out that using different types of linguistic term setsdoes not influence the theoretical derivation of distance and similarity measures for HFLTSs. The only difference takes placein their distinct semantics.

H. Liao et al. / Information Sciences 271 (2014) 125–142 129

Let S ¼ fsaja ¼ �s; . . . ;�1;0;1; . . . ; sg be a linguistic term set. The HFLTS HS for a linguistic variable v 2 V can then berepresented mathematically as HSðvÞ. For the convenience of statement, we call H ¼ fHSðvÞjv 2 Vg a set of HFLTSs. The aimof introducing HFLTS is to improve the elicitation of linguistic information, mainly when decision makers hesitate betweenseveral values in assessing linguistic variables. Linguistic information, which is more similar to the decision makers’expressions, is semantically represented by HFLTS and generated by a context-free grammar [26].

Example 1. Let S ¼ fs0 ¼ none; s1 ¼ very low; s2 ¼ low; s3 ¼ medium; s4 ¼ high; s5 ¼ very high; s6 ¼ perfectg be a linguis-tic term set. The linguistic information obtained by means of the context-free grammar might be/1 ¼ high; /2 ¼ lower than medium;/3 ¼ greater than high, and /4 ¼ between medium and very high. Then, according to

the definition of HFLTS, the above linguistic information can be represented as HS ¼ H1S ;H

2S ;H

3S ;H

4S

n owith

H1S ¼ fs4g; H2

S ¼ fs0; s1; s2; s3g; H3S ¼ fs4; s5; s6g and H4

S ¼ fs3; s4; s5g being four HFLTSs.Rodríguez et al. [26] defined the complement, union and intersection of HFLTSs, which are represented as follows:

Definition 4 [26]. For three HFLTSs HS; H1S and H2

S , the following operations are defined:

(1) Lower bound: H�S ¼ minðsiÞ ¼ sj; si 2 hS and si P sj; 8i.(2) Upper bound: HþS ¼maxðsiÞ ¼ sj; si 2 hS and si 6 sj; 8i.(3) Hc

S ¼ S� HS ¼ fsijsi 2 S and si R HSg.(4) H1

S [ H2S ¼ sijsi 2 H1

S or si 2 H2S

n o.

(5) H1S \ H2

S ¼ sijsi 2 H1S and si 2 H2

S

n o.

From Example 1, we can see that different HFLTSs have different numbers of linguistic terms in most cases. In order tooperate correctly when comparing two HFLTSs, Zhu and Xu [53] introduced a method to add linguistic terms in a HFLTS.Let b ¼ fbljl ¼ 1; . . . ;#bg be a HFLTS (#b is the number of linguistic terms in b), bþ and b� be the maximum and minimumlinguistic terms in b respectively, and nð0 6 n 6 1Þ be an optimized parameter. Then we can add the linguistic term

b��¼ nbþ � ð1� nÞb� ð3Þ

into the HFLTS. The max, min and the averaged linguistic terms correspond with b�; b�, and bA respectively, whereb� ¼ bþ; b� ¼ b�, and bA ¼ 1

2 ðbþ � b�Þ. It’s obvious that b� and b� correspond with the optimism and pessimism rules, respec-

tively. The optimized parameter is used to reflect decision makers’ risk preferences. Without loss of generality, in this paper,we assume n ¼ 1

2.

3. Distance and similarity measures between two HFLTSs

Distance and similarity measures are common tools used widely in measuring the deviation and closeness degrees of dif-ferent arguments. Up to now, many scholars have paid great attention to this issue and have achieved many results, whichcan be roughly classified into two sorts: one sort is based on the traditional distance measures, such as the Hamming dis-tance, the Euclidean distance, and the Hausdorff metric [7,11,12,28,36,43]. The other is on the basis of some weighted dis-tance operators, such as the ordered weighted distance measures [42], the hybrid weighted distance measures [39], and thefuzzy ordered distance measures [40]. All these distance measures have been extended into fuzzy sets [23,35,39,40,42],intuitionistic fuzzy sets [2,7,11,17,28,29,38,43,47], interval-valued intuitionistic fuzzy sets [1,2,7,47], linguistic fuzzy sets[36,37,41,45] and HFSs [13,14,46].

For the first sort of distance and similarity measures, within the context of intuitionistic fuzzy sets (IFSs), Burillo and Bust-ince [2] defined the normalized Hamming distance and the normalized Euclidean distance, which only involve the first twoparameters, the membership degree and the nonmembership degree in describing an IFS. Considering the geometrical rep-resentation of distances between IFSs, Szmidt and Kacprzyk [28] showed that the third parameter (the degree of indetermi-nacy) should not be omitted when calculating distances between IFSs. Then they defined some similarity measures for IFSsbased on the proposed distance measures [29]. Hung and Yang [11] presented a method to calculate the distances betweenIFSs on the basis of the Hausdorff distance and then defined the similarity measures for IFSs. Grzegorzewski [7] also proposedsome distance measures based on the Hausdorff metric. Interval-valued intuitionistic fuzzy sets (IVIFSs) [1] are a generalizedcase of IFSs whose values are intervals rather than exact numbers. Thus their distance and similarity measures can be de-rived in analogy to IFSs. Xu and Chen [43] gave a comprehensive overview of distance and similarity measures of IFSsand defined some continuous distance and similarity measures for IFSs and IVIFSs. Xu [37] defined the concepts of deviationdegree and similarity degree between two linguistic values, as well as the concepts of deviation degree and similarity degreebetween two linguistic preference relations. Xu and Xia [46] proposed a variety of distance measures for HFSs, based onwhich the corresponding similarity measures were obtained.

On the other hand, based on the weighting operators, distinct forms of distance and similarity measures have been pro-posed. Based on the Choquet integral [5] with respect to the non-monotonic fuzzy measure, Narukawa and Torra [23] intro-duced a weighted distance measure for IFSs. Based on the geometric distance model, Xu [38] generalized Szmidt and

130 H. Liao et al. / Information Sciences 271 (2014) 125–142

Kacprzyk [28]’s distance measures into the geometric forms and further defined some weighted distance measures for IFSs,based on which the similarity measures were also proposed. Xu and Wang [45] extended the distance measure to the lin-guistic fuzzy set, by developing several linguistic distance operators, such as the linguistic weighted distance operator,the linguistic ordered weighted distance operator, and studying some of their desired properties. Xu and Xia [46] developeda number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures for HFSs.

All the aforementioned measures can not be used to deal with the distance and similarity between two HFLTSs, but theygive good inspiration for us to develop the distance and similarity measures for HFLTSs. Motivated by the above analysis, wecan set out our investigation of the distance and similarity measures over HFLTSs from two aspects, i.e., the extensions oftraditional distance and similarity measures and the different types of weighted forms.

Let’s first put forward the axioms of distance and similarity measures for HFLTSs.

Definition 5. Let S ¼ fsaja ¼ �s; . . . ;�1;0;1; . . . ; sg be a linguistic term set, H1S and H2

S be two HFLTSs, then the distance

measure between H1S and H2

S is defined as d H1S ;H

2S

� �, which satisfies:

(1) 0 6 d H1S ;H

2S

� �6 1;

(2) d H1S ;H

2S

� �¼ 0 if and only if H1

S ¼ H2S ;

(3) d H1S ;H

2S

� �¼ d H2

S ;H1S

� �.

Definition 6. Let S ¼ fsaja ¼ �s; . . . ;�1;0;1; . . . ; sg be a linguistic term set, H1S and H2

S be two HFLTSs, then the similarity

measure between H1S and H2

S is defined as q H1S ;H

2S

� �, which satisfies:

(1) 0 6 q H1S ;H

2S

� �6 1;

(2) q H1S ;H

2S

� �¼ 1 if and only if H1

S ¼ H2S ;

(3) q H1S ;H

2S

� �¼ q H2

S ;H1S

� �.

The axioms defined here are similar to the axioms of distance and similarity measures for HFSs given by Xu and Xia [46].The three conditions are easy to be understood, and each of them is essential for the definition of the measures. In otherword, each different form of distance or similarity measure should satisfy these three conditions respectively. Noticing thatthe relationship between distance and similarity measures is

q H1S ;H

2S

� �¼ 1� d H1

S ;H2S

� �ð4Þ

in this paper, we mainly discuss the distance measures of HFLTSs, and the corresponding similarity measures can be ob-tained easily by using Eq. (4).

In Ref. [37], Xu gave the definition of distance measure between any two linguistic terms:

Definition 7 [37]. Let S ¼ fsaja ¼ �s; . . . ;�1;0;1; . . . ; sg be a linguistic term set, sa; sb 2 S be two linguistic terms, then thedistance measure between sa and sb is

dðsa; sbÞ ¼ja� bj2sþ 1

ð5Þ

where 2sþ 1 is the number of linguistic terms in the set S.Let S ¼ fsaja ¼ �s; . . . ;�1;0;1; . . . ; sg be a linguistic term set, H1

S ðxiÞ ¼ [sd1l2H1

Ssd1

ljl ¼ 1; . . . ;#H1

S

n o(#H1

S be the number of

linguistic terms in H1S ) and H2

S ðxiÞ ¼ [sd2l2H2

Ssd2

ljl ¼ 1; . . . ;#H2

S

n obe two HFLTSs on X ¼ fx1; x2; . . . ; xng, where #H1

S ¼ #H2S ¼ L

(otherwise, we can extend the shorter one by adding the linguistic terms given as Eq. (3)). Suppose that the linguistic terms

are arranged in ascending order, motivated by Definition 5, we can introduce the Hamming distance and the Euclidean dis-

tance for HFLTSs. The Hamming distance of H1S ðxiÞ and H2

S ðxiÞ can be defined as:

dhd H1S ðxiÞ;H2

S ðxiÞ� �

¼ 1L

XL

l¼1

d1l � d2

l

�� ��2sþ 1

ð6Þ

and the Euclidean distance of H1S ðxiÞ and H2

S ðxiÞ can be defined as:

ded H1S ðxiÞ;H2

S ðxiÞ� �

¼ 1L

XL

l¼1

d1l � d2

l

�� ��2sþ 1

!20@

1A

1=2

ð7Þ

H. Liao et al. / Information Sciences 271 (2014) 125–142 131

For the Hamming distance defined as Eq. (6), since �s 6 d1l 6 s and �s 6 d2

l 6 s, then we have 0 6 d1l � d2

l

�� �� 6 2s. Thus,

0 6 dhd H1S ;H

2S

� �6 1. In addition, if H1

S ¼ H2S , i.e., d1

l ¼ d2l , for l ¼ 1; . . . ; L, then dhd H1

S ;H2S

� �¼ 0. On the other hand, if

dhd H1S ;H

2S

� �¼ 0, then we can derive that H1

S ¼ H2S . It is obvious that dhd H1

S ;H2S

� �¼ dhd H2

S ;H1S

� �. Hence, the Hamming distance

measure satisfies the axioms of distance measure for HFLTSs given as Definition 5. All the other distance measures can beverified easily in a similar way and thus we omit the details here.

Motivated by the generalized idea by Yager [49], we can unify the Hamming distance and the Euclidean distance by thefollowing generalized distance measure:

dgd H1S ðxiÞ;H2

S ðxiÞ� �

¼ 1L

XL

l¼1

d1l � d2

l

�� ��2sþ 1

!k0@

1A

1=k

ð8Þ

where k > 0. In particular, if k ¼ 1, then the above generalized distance becomes the Hamming distance; If k ¼ 2, then theabove generalized distance becomes the Euclidean distance.

Example 2. Let S ¼ fsaja ¼ �3; . . . ;�1;0;1; . . . ;3g be a linguistic term set, H1S ¼ fs1; s2g and H2

S ¼ fs�3; s�1; s3g be two HFLTSson S. Further we can extend H1

S to H1S ¼ fs1; s1:5; s2g by adding the linguistic term s1:5. Thus, the generalized distance between

H1S and H2

S is

ded H1S ;H

2S

� �¼ 1

3j1� ð�3Þj

7

� �k

þ j1:5� ð�1Þj7

� �k

þ j2� 3j7

� �k ! !1=k

If k ¼ 1, then the Hamming distance between H1S and H2

S is

dhd H1S ;H

2S

� �¼ 1

3j1� ð�3Þj

7þ j1:5� ð�1Þj

7þ j2� 3j

7

� �¼ 0:3571

If k ¼ 2, the Euclidean distance of H1S and H2

S is

dedðH1S ;H

2S Þ ¼

13

j1� ð�3Þj7

� �2

þ j1:5� ð�1Þj7

� �2

þ j2� 3j7

� �2 ! !1=2

¼ 0:3977

Similarly, the Hausdorff distance measure can be introduced to HFLTS. For two HFLTSs H1S ðxiÞ and H2

S ðxiÞ, the generalizedHausdorff distance measure can be defined as:

dghaud H1S ðxiÞ;H2

S ðxiÞ� �

¼ maxl¼1;2;...;L

d1l � d2

l

�� ��2sþ 1

!k0@

1A

1=k

ð9Þ

where k > 0.In particular, if k ¼ 1, then the above generalized Hausdorff distance becomes the Hamming–Hausdorff distance:

dhhaud H1S ðxiÞ;H2

S ðxiÞ� �

¼ maxl¼1;2;...;L

d1l � d2

l

�� ��2sþ 1

ð10Þ

If k ¼ 2, then the above generalized Hausdorff distance becomes the Euclidean–Hausdorff distance:

dehaud H1S ðxiÞ;H2

S ðxiÞ� �

¼ maxl¼1;2;...;L

d1l � d2

l

�� ��2sþ 1

!20@

1A

1=2

ð11Þ

In addition, we can obtain some hybrid distance measures via combining the above distances measures, such as:

(1) The hybrid Hamming distance between H1S ðxiÞ and H2

S ðxiÞ:

dhhd H1S ðxiÞ;H2

S ðxiÞ� �

¼ 12

1L

XL

l¼1

d1l � d2

l

�� ��2sþ 1

þ maxl¼1;2;...;L

d1l � d2

l

�� ��2sþ 1

!ð12Þ

(2) The hybrid Euclidean distance between H1S ðxiÞ and H2

S ðxiÞ:

dhed H1S ðxiÞ;H2

S ðxiÞ� �

¼ 12

1L

XL

l¼1

d1l � d2

l

�� ��2sþ 1

!2

þ maxl¼1;2;...;L

d1l � d2

l

�� ��2sþ 1

!20@

1A

0@

1A

1=2

ð13Þ

132 H. Liao et al. / Information Sciences 271 (2014) 125–142

(3) The generalized hybrid distance between H1S ðxiÞ and H2

S ðxiÞ:

dghd H1S ðxiÞ;H2

S ðxiÞ� �

¼ 12

1L

XL

l¼1

d1l � d2

l

�� ��2sþ 1

!k

þ maxl¼1;2;...;L

d1l � d2

l

�� ��2sþ 1

!k0@

1A

0@

1A

1=k

ð14Þ

where k > 0.

Example 3 (Continue with Example 2). According to Eq. (14), the generalized Hausdorff distance between H1S and H2

S is

dghaudðH1S ;H

2S Þ ¼ max

j1� ð�3Þj7

� �k

;j1:5� ð�1Þj

7

� �k

;j2� 3j

7

� �k( ) !1=k

Then, if k ¼ 1, then the Hamming–Hausdorff distance between H1S and H2

S is

dhhaud H1S ;H

2S

� �¼max

j1� ð�3Þj7

;j1:5� ð�1Þj

7;j2� 3j

7

� �¼ 0:5714

if k ¼ 2, then the Euclidean–Hausdorff distance between H1S and H2

S is

dehaud H1S ;H

2S

� �¼ max

j1� ð�3Þj7

� �2

;j1:5� ð�1Þj

7

� �2

;j2� 3j

7

� �2( ) !1=2

¼ 0:5714

The generalized hybrid distance between H1S and H2

S is

dghd H1S ;H

2S

� �¼ 1

213

47

� �k

þ 2:57

� �k

þ 17

� �k !

þmax47

� �k

;2:57

� �k

;17

� �k( ) ! !1=k

If k ¼ 1, then the hybrid Hamming distance between H1S and H2

S is

dhhd H1S ;H

2S

� �¼ 1

20:3571þ 0:5714ð Þ ¼ 0:4643

If k ¼ 2, then the hybrid Euclidean distance between H1S and H2

S is

dhed H1S ;H

2S

� �¼ 1

2ð0:1582þ 0:3265Þ

� �1=2

¼ 0:4733

4. Distance and similarity measures between two collections of HFLTSs

In the aforementioned section, we actually only consider the distance and similarity measures of HFLTSs over one singlelinguistic variable. However, in many real applications such as multi-criteria decision making, the objects/alternatives areoften evaluated with respect to different attributes/criteria. Hence, we need to take all the aspects into account. In addition,the weighting information of different criteria is very important and thus has to be considered. When the evaluation infor-mation of alternatives with respect to different criteria is represented by several collections of HFLTSs, we need to calculatethe distance and similarity measures between these collections of HFLTSs in order to compare the considered alternatives. Inthis section, we mainly focus on the weighted distance measures for two collections of HFLTSs.

4.1. Distance and similarity measures between two collections of HFLTSs in discrete case

Let S ¼ fsaja ¼ �s; . . . ;�1;0;1; . . . ; sg be a linguistic term set. For two collections of HFLTSs H1S ¼ H11

S ;H12S ; . . . ;H1m

S

n oand

H2S ¼ H21

S ;H22S ; . . . ;H2m

S

n owith the associated weighting vector w ¼ ðw1;w2; . . . ;wmÞT , where 0 6 wj 6 1 and

Pmj¼1 wj ¼ 1, a

generalized weighted distance measure between H1S and H2

S is defined as:

dgwd H1S ;H

2S

¼

Xm

j¼1

wj

L

XL

l¼1

d1jl � d2j

l

��� ���2sþ 1

0@

1A

k0B@

1CA

1=k

ð15Þ

and a generalized weighted Hausdorff distance measure between H1S and H2

S is defined as:

dgwhaud H1S ;H

2S

¼

Xm

j¼1

wj maxl¼1;2;...;L

d1jl � d2j

l

��� ���2sþ 1

0@

1A

k0B@

1CA

1=k

ð16Þ

H. Liao et al. / Information Sciences 271 (2014) 125–142 133

where k > 0.In particular, if k ¼ 1, then we obtain the weighted Hamming distance between H1

S and H2S :

dwhd H1S ;H

2S

¼Xm

j¼1

wj

L

XL

l¼1

d1jl � d2j

l

��� ���2sþ 1

ð17Þ

and the weighted Hamming–Hausdorff distance between H1S and H2

S :

dwhhaud H1S ;H

2S

¼Xm

j¼1

wj maxl¼1;2;...;L

d1jl � d2j

l

��� ���2sþ 1

ð18Þ

If k ¼ 2, then we can get the weighted Euclidean distance between H1S and H2

S :

dwed H1S ;H

2S

¼

Xm

j¼1

wj

L

XL

l¼1

d1jl � d2j

l

��� ���2sþ 1

0@

1A

20B@

1CA

1=2

ð19Þ

and the weighted Euclidean–Hausdorff distance:

dwehaud H1S ;H

2S

¼

Xm

j¼1

wj maxl¼1;2;...;L

d1jl � d2j

l

��� ���2sþ 1

0@

1A

20B@

1CA

1=2

ð20Þ

Certainly, we can derive some hybrid weighted distance measures via combining the above distance measures, such as:

(1) The hybrid weighted Hamming distance between H1S and H2

S :

dhwhd H1S ;H

2S

¼Xm

j¼1

wj

21L

XL

l¼1

d1jl � d2j

l

��� ���2sþ 1

þ maxl¼1;2;...;L

d1jl � d2j

l

��� ���2sþ 1

0@

1A ð21Þ

(2) The hybrid weighted Euclidean distance between H1S and H2

S :

dhwed H1S ;H

2S

¼

Xm

j¼1

wj

21L

XL

l¼1

d1jl � d2j

l

��� ���2sþ 1

0@

1A

2

þ maxl¼1;2;...;L

d1jl � d2j

l

��� ���2sþ 1

0@

1A

20B@

1CA

0B@

1CA

1=2

ð22Þ

(3) The generalized hybrid weighted distance between H1S and H2

S :

dghwd H1S ;H

2S

¼

Xm

j¼1

wj

21L

XL

l¼1

d1jl � d2j

l

��� ���2sþ 1

0@

1A

k

þ xl¼1;2;...;L

d1jl � d2j

l

��� ���2sþ 1

0@

1A

k0B@

1CA

0B@

1CA

1=k

ð23Þ

where k > 0.

4.2. Distance and similarity measures between two collections of HFLTSs in continuous case

In the last subsection, all the considered distance measures are based on discrete input data. However, sometimes theuniverse of discourse and the weights of elements are continuous. This subsection focuses on this case.

Let x 2 ½a; b�, and the weight of x be wðxÞ, where wðxÞ 2 ½0;1� andR b

a wðxÞdx ¼ 1. Let H1S and H2

S be two collections of HFLTSsover the element x. Then, in analogy to the above analysis, we can introduce a continuous weighted Hamming distance mea-sure, a continuous weighted Euclidean distance measure, and a generalized continuous weighted distance measure betweentwo collections of HFLTSs H1

S and H2S , which are shown as follows, respectively:

dcwhd H1S ;H

2S

¼Z b

awðxÞ1

L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1dx ð24Þ

dcwed H1S ;H

2S

¼

Z b

awðxÞ1

L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!2

dx

0@

1A

1=2

ð25Þ

dgcwd H1S ;H

2S

¼

Z b

awðxÞ1

L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!k

dx

0@

1A

1=k

ð26Þ

134 H. Liao et al. / Information Sciences 271 (2014) 125–142

where k > 0.If wðxÞ ¼ 1=ðb� aÞ; 8 x 2 ½a; b�, then Eqs. (24)–(26) reduce to a continuous normalized Hamming distance measure, a con-

tinuous normalized Euclidean distance measure and a generalized continuous normalized distance measure between twocollections of HFLTSs respectively, which are shown as follows:

dcnhd H1S ;H

2S

¼ 1

b� a

Z b

a

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1dx ð27Þ

dcned H1S ;H

2S

¼ 1

b� a

Z b

a

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!2

dx

0@

1A

1=2

ð28Þ

dgcnd H1S ;H

2S

¼ 1

b� a

Z b

a

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!k

dx

0@

1A

1=k

ð29Þ

where k > 0.Now we consider the Hausdorff metric. Similar to the above, a generalized continuous weighted distance measure, a con-

tinuous weighted Hamming–Hausdorff distance measure and a continuous weighted Euclidean–Hausdorff distance measurebetween two collections of HFLTSs H1

S and H2S can be obtained as follows:

dgcwhaud H1S ;H

2S

¼

Z b

awðxÞ max

l¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!k

dx

0@

1A

1=k

ð30Þ

dcwhhaud H1S ;H

2S

¼Z b

awðxÞ max

l¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1dx ð31Þ

dcwehaud H1S ;H

2S

¼

Z b

awðxÞ max

l¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!2

dx

0@

1A

1=2

ð32Þ

If wðxÞ ¼ 1=ðb� aÞ; 8 x 2 ½a; b�, then Eqs. (30)–(32) reduce to a generalized continuous normalized distance measure, acontinuous normalized Hamming–Hausdorff distance measure and a continuous normalized Euclidean–Hausdorff distancemeasure between two collections of HFLTSs H1

S and H2S respectively, which can be shown as follows:

dgcwhaud H1S ;H

2S

¼ 1

b� a

Z b

amax

l¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!k

dx

0@

1A

1=k

ð33Þ

dcwhhaud H1S ;H

2S

¼ 1

b� a

Z b

amax

l¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1dx ð34Þ

dcwehaud H1S ;H

2S

¼ 1

b� a

Z b

amax

l¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!2

dx

0@

1A

1=2

ð35Þ

where k > 0.Naturally, we can derive some hybrid continuous weighted distance measures, such as a hybrid continuous weighted

Hamming distance measure, a hybrid continuous weighted Euclidean distance measure and a generalized hybrid continuousweighted distance between two collections of HFLTSs H1

S and H2S , which are shown below:

dhcwhd H1S ;H

2S

¼Z b

a

wðxÞ2

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1þ max

l¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!dx ð36Þ

dhcwed H1S ;H

2S

¼

Z b

a

wðxÞ2

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!2

þ maxl¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!20@

1Adx

0@

1A

1=2

ð37Þ

dghcwd H1S ;H

2S

¼

Z b

a

wðxÞ2

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!k

þ maxl¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!k0@

1Adx

0@

1A

1=k

ð38Þ

where k > 0.Let wðxÞ ¼ 1=ðb� aÞ; 8 x 2 ½a; b�, then, Eqs. (36)–(38) reduce to a hybrid continuous normalized Hamming distance mea-

sure, a hybrid continuous normalized Euclidean distance measure and a generalized hybrid continuous normalized distancebetween two collections of HFLTSs H1

S and H2S respectively:

H. Liao et al. / Information Sciences 271 (2014) 125–142 135

dhcwhd H1S ;H

2S

¼ 1

2ðb� aÞ

Z b

a

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1þ max

l¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!dx ð39Þ

dhcwed H1S ;H

2S

¼ 1

2ðb� aÞ

Z b

a

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!2

þ maxl¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!20@

1Adx

0@

1A

1=2

ð40Þ

dghcwd H1S ;H

2S

¼ 1

2ðb� aÞ

Z b

a

1L

XL

l¼1

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!k

þ maxl¼1;2;...;L

d1l ðxÞ � d2

l ðxÞ�� ��

2sþ 1

!k0@

1Adx

0@

1A

1=k

ð41Þ

where k > 0.

4.3. Ordered weighted distance and similarity measures between two collections of HFLTSs

Ordered weighted distance and similarity measures, first proposed by Xu and Chen [42], have also been investigated bymany scholars and have been applied to fuzzy sets [40], linguistic fuzzy sets [40,41,45] and HFSs [46]. The prominent char-acteristic of ordered weighted distance is that it can alleviate/intensify the influence of unduly large/small deviations on theaggregation results by assigning them low/high weights. This desired property makes the ordered weighted distance mea-sure very useful in realistic decision making problems. In [40], Xu considered the situations with linguistic, interval or fuzzypreference information and developed some fuzzy ordered distance measures. Xu and Wang [45] developed the linguisticordered weighted distance operator and studied its properties, such as commutativity, monotonicity, idempotency andboundedness. Within the context of HFSs, Xu and Xia [46] proposed a variety of ordered weighted distance measures. Yager[50] generalized Xu and Chen’s distance measures in [42] and introduced a collection of ordered weighted averaging norms.Below we will pay our attention to the ordered weighted distance measures within the context of HFLTSs.

Inspired by Xu and Chen [42], we can introduce a generalized ordered weighted distance between two collections ofHFLTSs H1

S and H2S :

dgowd H1S ;H

2S

¼

Xm

j¼1

wj1L

XL

l¼1

d1rðjÞl � d2rðjÞ

l

��� ���2sþ 1

0@

1A

k0B@

1CA

0B@

1CA

1=k

ð42Þ

where k > 0 and rðjÞ : ð1;2; . . . ;mÞ ! ð1;2; . . . ;mÞ is a permutation such that

1L

XL

l¼1

d1rðjÞl � d2rðjÞ

l

��� ���2sþ 1

0@

1A

k

P1L

XL

l¼1

d1rðjþ1Þl � d2rðjþ1Þ

l

��� ���2sþ 1

0@

1A

k

; j ¼ 1;2; . . . ;m ð43Þ

In particular, if k ¼ 1, then the generalized ordered weighted distance becomes the ordered weighted Hamming distancebetween H1

S and H2S :

dowhd H1S ;H

2S

¼Xm

j¼1

wj

L

XL

l¼1

d1rðjÞl � d2rðjÞ

l

��� ���2sþ 1

ð44Þ

If k ¼ 2, then the generalized ordered weighted distance becomes the ordered weighted Euclidean distance between H1S and

H2S :

dowed H1S ;H

2S

¼

Xm

j¼1

wj1L

XL

l¼1

d1rðjÞl � d2rðjÞ

l

��� ���2sþ 1

0@

1A

20B@

1CA

0B@

1CA

1=2

ð45Þ

Based on the Hausdorff metric, we can also develop a generalized ordered weighted Hausdorff distance between two col-lections of HFLTSs H1

S and H2S :

dgowhaud H1S ;H

2S

¼

Xm

j¼1

wj maxl¼1;2;...;L

d1 _rðjÞl � d2 _rðjÞ

l

��� ���2sþ 1

0@

1A

k0B@

1CA

1=k

ð46Þ

where k > 0 and _rðjÞ : ð1;2; . . . ;mÞ ! ð1;2; . . . ;mÞ is a permutation such that

maxl¼1;2;...;L

d1 _rðjÞl � d2 _rðjÞ

l

��� ���2sþ 1

0@

1A

k

P maxl¼1;2;...;L

d1 _rðjþ1Þl � d2 _rðjþ1Þ

l

��� ���2sþ 1

0@

1A

k

; j ¼ 1;2; . . . ;m ð47Þ

136 H. Liao et al. / Information Sciences 271 (2014) 125–142

If k ¼ 1, then the generalized ordered weighted Hausdorff distance becomes the ordered weighted Hamming–Hausdorffdistance between H1

S and H2S :

dowhhaud H1S ;H

2S

¼Xm

j¼1

wj maxl¼1;2;...;L

d1 _rðjÞl � d2 _rðjÞ

l

��� ���2sþ 1

ð48Þ

If k ¼ 2, then the generalized ordered weighted Hausdorff distance becomes the ordered weighted Euclidean–Hausdorffdistance:

dowehaud H1S ;H

2S

¼

Xm

j¼1

wj maxl¼1;2;...;L

d1 _rðjÞl � d2 _rðjÞ

l

��� ���2sþ 1

0@

1A

20B@

1CA

1=2

ð49Þ

Certainly, we can also derive some hybrid ordered weighted distance measures via combining the above distance mea-sures, such as:

(1) The hybrid ordered weighted Hamming distance between H1S and H2

S :

dhwhd H1S ;H

2S

¼Xm

j¼1

wj

21L

XL

l¼1

d1€rðjÞl � d2€rðjÞ

l

��� ���2sþ 1

þ maxl¼1;2;...;L

d1€rðjÞl � d2€rðjÞ

l

��� ���2sþ 1

0@

1A ð50Þ

(2) The hybrid ordered weighted Euclidean distance between H1S and H2

S :

dhwed H1S ;H

2S

¼

Xm

j¼1

wj

21L

XL

l¼1

d1€rðjÞl � d2€rðjÞ

l

��� ���2sþ 1

0@

1A

2

þ maxl¼1;2;...;L

d1€rðjÞl � d2€rðjÞ

l

��� ���2sþ 1

0@

1A

20B@

1CA

0B@

1CA

1=2

ð51Þ

(3) The generalized hybrid ordered weighted distance between H1S and H2

S :

dghwd H1S ;H

2S

¼

Xm

j¼1

wj

21L

XL

l¼1

d1€rðjÞl � d2€rðjÞ

l

��� ���2sþ 1

0@

1A

k

þ maxl¼1;2;...;L

d1€rðjÞl � d2€rðjÞ

l

��� ���2sþ 1

0@

1A

k0B@

1CA

0B@

1CA

1=k

ð52Þ

where k > 0, and €rðjÞ : ð1;2; . . . ;mÞ ! ð1;2; . . . ;mÞ is a permutation such that

1L

XL

l¼1

d1€rðjÞl � d2€rðjÞ

l

��� ���2sþ 1

0@

1A

2

þ maxl¼1;2;...;L

d1€rðjÞl � d2€rðjÞ

l

��� ���2sþ 1

0@

1A

2

P1L

XL

l¼1

d1€rðjþ1Þl � d2€rðjþ1Þ

l

��� ���2sþ 1

0@

1A

2

þ maxl¼1;2;...;L

d1€rðjþ1Þl � d2€rðjþ1Þ

l

��� ���2sþ 1

0@

1A

2

;

j ¼ 1;2; . . . ;m ð53Þ

5. An approach based on distance and similarity measures to multi-criteria decision making with HFLTSs

Multi-criteria decision making, which can be characterized in terms of a process of choosing or selecting sufficiently goodalternative(s) (or course(s)) from a set of alternatives to attain a goal (or goals), often happens in our daily life. For example,choosing a car to buy, or selecting an electronic product from amazon or ebay [17,44]. A multi-criteria decision making prob-lem with HFLTS information can be interpreted as follows: Suppose that a decision maker is asked to evaluate a set of alter-natives X ¼ fx1; x2; . . . ; xng with respect to several criteria cjðj ¼ 1;2; . . . ;mÞ. The criteria have a weighting vector

w ¼ ðw1;w2; . . . ;wmÞT , where 0 6 wj 6 1 andPm

j¼1 wj ¼ 1. The decision maker might feel much easier and are more willingto give their assessments by providing some linguistic expressions or sentences. Such linguistic information can be trans-formed into HFLTSs by using the context-free grammar [26]. Hence, a judgment matrix with HFLTS information will be ob-tained as follows:

HS ¼

H11S H12

S � � � H1mS

H21S H22

S � � � H2mS

..

. ... . .

. ...

Hn1S Hn2

S � � � HnmS

2666664

3777775 ð54Þ

where HijS ¼

Ssdijl

2HijSfs

dijljl ¼ 1; . . . ;#Hij

Sg ði ¼ 1;2; . . . ;n; j ¼ 1;2; . . . mÞ is a HFLTS, denoting the degree that the alternative xi

satisfies the criterion cj.

H. Liao et al. / Information Sciences 271 (2014) 125–142 137

For each HFLTS HijS , according to Definition 3, we can obtain the lower bound Hij�

S ¼minl¼1;...;#HijSfs

dijlg and the upper bound

HijþS ¼maxl¼1;...;#Hij

Sfs

dijlg. Then, we can define the notions of the hesitant fuzzy linguistic positive ideal solution xþ and hesitant

fuzzy linguistic negative ideal solution x� as follows, respectively:

xþ ¼ H1þS ;H2þ

S ; . . . ;HmþS

n oð55Þ

and

x� ¼ H1�S ;H2�

S ; . . . ;Hm�S

n oð56Þ

where

HjþS ¼

maxi¼1;2;...;n

HijþS ¼ max

i¼1;2;...;n

l¼1;...;#HijS

fsdij

lg for benefit criterion cj

mini¼1;2;...;n

Hij�S ¼ min

i¼1;2;...;n

l¼1;...;#HijS

fsdij

lg for cos t criterion cj

8>>>><>>>>:

; for j ¼ 1;2; . . . ;m ð57Þ

and

Hj�S ¼

mini¼1;2;...;n

Hij�S ¼ min

i¼1;2;...;n

l¼1;...;#HijS

fsdij

lg for benefit criterion cj

maxi¼1;2;...;n

HijþS ¼ max

i¼1;2;...;n

l¼1;...;#HijS

fsdij

lg for cos t criterion cj

8>>>><>>>>:

; for j ¼ 1;2; . . . ;m ð58Þ

Note that the hesitant fuzzy linguistic positive ideal solution xþ and the hesitant fuzzy linguistic negative ideal solution x�

are linguistic term sets. Hence, they certainly can be taken as special HFLTSs with only one linguistic term in each HFLTS.In order to choose the desired alternative, we can calculate the distance between each alternative xi and the hesitant fuz-

zy linguistic positive ideal solution xþ, and the distance between each alternative xi and the hesitant fuzzy linguistic negativeideal solution x�, respectively. Intuitively, the smaller the distance dðxi; xþÞ, the better the alternative; while the larger thedistance dðxi; x�Þ, the better the alternative. All the proposed distance measures in Section 4 can be used to define and cal-culate these distances. Motivated by the well known TOPSIS (Technique for Order Preference by Similarity to an Ideal Solu-tion) method [3,13], we take both distance dðxi; xþÞ and dðxi; x�Þ into consideration simultaneously rather than separately.This leads naturally to the following concept of satisfaction degree:

Definition 8. A satisfaction degree of a given alternative xi over the criteria cjðj ¼ 1;2; . . . ;nÞ is defined as:

gðxiÞ ¼ð1� hÞdðxi; x�Þ

hdðxi; xþÞ þ ð1� hÞdðxi; x�Þð59Þ

where the parameter h denotes the risk preferences of the decision maker: h > 0:5 means that the decision maker is pessi-mists; while h < 0:5 means the opposite. The value of the parameter h is provided by the decision maker in advance.

It is obvious that 0 6 gðxiÞ 6 1, for any h 2 ½0;1�; i ¼ 1;2; . . . ;m. The higher the satisfaction degree, the better thealternative. Different distance measures proposed in the previous sections can be employed to define and calculate thesatisfaction degrees. In the following, we consider a multi-criteria decision making problem that concerns the evaluation ofthe quality of movies to illustrate our approach and the differences between different distance measures:

Example 4. Consider a movie recommender system. Suppose that a company intends to give ratings on five moviesm1;m2; . . . ;m5 with respect to four criteria: story ðf1Þ, acting (f2Þ, visuals ðf3Þ and direction ðf4Þ. The weighing vector of thesefour criteria is w ¼ ð0:4;0:2; 0:2;0:2ÞT . The ratings provide information about the quality of the movies as well as the taste ofthe users who give the ratings. Since these criteria are all qualitative, it is convenient and only feasible for the decision mak-ers to express their feelings by using linguistic terms. As pointed out by Miller [22], most decision makers cannot handlemore than 9 factors when making their decision. Hence, the company constructs a seven point linguistic scale to assessthe movies, which is S ¼ fs�3 ¼ terrible; s�2 ¼ very bad; s�1 ¼ bad; s0 ¼ medium; s1 ¼ well; s2 ¼ very well; s3 ¼ perfectg.

To get more objective and reasonable evaluation results, the company sets up a decision organization, which contains agroup of decision makers, to assess the movies. In the process of evaluation, the decision makers may think several linguisticterms at the same time for a movie over a criterion. For example, the decision makers may consider that the acting of themovie m2 is between medium and perfect. Such a linguistic expression is common and more similar to human being’scognition than just using a single linguistic term. The linguistic expression presented above is appropriate to be representedas a HFLTS fs0; s1; s2; s3g. In addition, the decision makers in the decision organization sometimes may have differentopinions on the movies, and sometimes they cannot reach some consensus results. For example, one decision maker maythink the direction of the movie m2 is perfect (denoted as s3Þ, and another person may think it is between medium and very

Table 1The hesitant fuzzy linguistic judgment matrix provided by the decision organization.

f1 f2 f3 f4

m1 fs�2; s�1; s0g fs0; s1g fs0 ; s1; s2g fs1; s2gm2 fs0; s1; s2g fs1; s2g fs0 ; s1g fs0; s1; s2gm3 fs2; s3g fs1; s2; s3g fs1; s2g fs2gm4 fs0; s1; s2g fs�1; s0; s1g fs1; s2; s3g fs1; s2gm5 fs�1; s0g fs0; s1; s2g fs0 ; s1; s2g fs0; s1g

Table 2The satisfaction degrees obtained by the generalized weighted distance measure.

m1 m2 m3 m4 m5 Rankings

k ¼ 1 0.3158 0.5263 0.8158 0.5526 0.3421 m3 � m4 � m2 � m5 � m1

k ¼ 2 0.3204 0.5420 0.7689 0.5357 0.3618 m3 � m2 � m4 � m5 � m1

k ¼ 4 0.3177 0.5509 0.7413 0.5250 0.3778 m3 � m2 � m4 � m5 � m1

k ¼ 6 0.3136 0.5549 0.7298 0.5206 0.3882 m3 � m2 � m4 � m5 � m1

k ¼ 10 0.3062 0.5593 0.7207 0.5154 0.4017 m3 � m2 � m4 � m5 � m1

Table 3The satisfaction degrees obtained by the generalized weighted Hausdorff distance measure.

m1 m2 m3 m4 m5 Rankings

k ¼ 1 0.3704 0.5185 0.7500 0.5357 0.3846 m3 � m4 � m2 � m5 � m1

k ¼ 2 0.3500 0.5341 0.7312 0.5285 0.3896 m3 � m2 � m4 � m5 � m1

k ¼ 4 0.3279 0.5496 0.7217 0.5226 0.3972 m3 � m2 � m4 � m5 � m1

k ¼ 6 0.3162 0.5563 0.7181 0.5198 0.4031 m3 � m2 � m4 � m5 � m1

k ¼ 10 0.3046 0.5619 0.7153 0.5154 0.4116 m3 � m2 � m4 � m5 � m1

Table 4The satisfaction degrees obtained by the generalized hybrid weighted distance measure.

m1 m2 m3 m4 m5 Rankings

k ¼ 1 0.3516 0.5165 0.7976 0.5484 0.3596 m3 � m4 � m2 � m5 � m1

k ¼ 2 0.3395 0.5362 0.7526 0.5321 0.3773 m3 � m2 � m4 � m5 � m1

k ¼ 4 0.3246 0.5500 0.7298 0.5234 0.3905 m3 � m2 � m4 � m5 � m1

k ¼ 6 0.3154 0.5559 0.7222 0.5201 0.3984 m3 � m2 � m4 � m5 � m1

k ¼ 10 0.3051 0.5612 0.7171 0.5154 0.4086 m3 � m2 � m4 � m5 � m1

1 2 3 4 50.2

0.3

0.4

0.5

0.6

0.7

0.8

mj,j=1,2,3,4,5

Satis

fact

ion

degr

ees

λ=1λ=2λ=4λ=6λ=10

Fig. 4. The satisfaction degrees obtained by the generalized weighted distance measure.

138 H. Liao et al. / Information Sciences 271 (2014) 125–142

1 2 3 4 50.2

0.3

0.4

0.5

0.6

0.7

0.8

mj,j=1,2,3,4,5

Satis

fact

ion

degr

ees

λ=1λ=2λ=4λ=6λ=10

Fig. 5. The satisfaction degrees obtained by the generalized weighted Hausdorff distance measure.

1 2 3 4 50.2

0.3

0.4

0.5

0.6

0.7

0.8

mj,j=1,2,3,4,5

Satis

fact

ion

degr

ees

λ=1λ=2λ=4λ=6λ=10

Fig. 6. The satisfaction degrees obtained by the generalized hybrid weighted distance measure.

1 2 4 6 100.2

0.3

0.4

0.5

0.6

0.7

0.8 m1

m2

m3

m4

m5

Satis

fact

ion

degr

ees

λ

Fig. 7. The satisfaction degrees obtained by the generalized weighted Hausdorff distance measure with different parameter values.

H. Liao et al. / Information Sciences 271 (2014) 125–142 139

140 H. Liao et al. / Information Sciences 271 (2014) 125–142

well ðfs0; s1; s2gÞ. If they cannot persuade each other, then we can represent the assessment as a HFLTS fs0; s1; s2; s3g. Afterdiscussion, the final assessments of these five movies can be established and a hesitant fuzzy linguistic judgment matrix canbe constructed, shown as Table 1.

In order to select the desired movie, we first need to establish the hesitant fuzzy linguistic positive ideal solution mþ andthe hesitant fuzzy linguistic negative ideal solution m�, which can be conducted easily via (55)–(58) and shown asmþ ¼ ffs3g; fs3g; fs3g; fs2gg and m� ¼ ffs�2g; fs�1g; fs0g; fs0gg. It is noted that all the four criteria are benefit-type criteria.Then we can calculate the distance between each alternative mi and the hesitant fuzzy linguistic positive ideal solution mþ,and the distance between each alternative mi and the hesitant fuzzy linguistic negative ideal solution m�, respectively.Furthermore, the satisfaction degree gðmiÞ for each movie mi can be calculated using Eq. (59). Without loss of generality, wechoose h ¼ 0:5.

If we use the generalized weighted distance measure, the generalized weighted Hausdorff distance measure, and thegeneralized hybrid weighted distance measure to calculate the distances, then the satisfaction degrees will be different,shown as Tables 2–4 and also Figs. 4–9, respectively.

From Tables 2–4, we can see that the rankings are the same when using different distance measures. But as the parameterk changes, the rankings change slightly: when k ¼ 1, the ranking is m3 � m4 � m2 � m5 � m1; but when k ¼ 2;4;6;10,slightly change takes places between m2 and m4. All of the results show that m3 is the best alternative, which means the thirdmovie is the best choice for the company. Such a conclusion can be drawn directly from Figs. 4–9, in which m3 is always atthe top of the figures.

1 2 4 6 100.2

0.3

0.4

0.5

0.6

0.7

0.8

λ

Satis

fact

ion

degr

ees

m1

m2

m3

m4

m5

Fig. 8. The satisfaction degrees obtained by the generalized weighted distance measure with different parameter values.

1 2 4 6 100.2

0.3

0.4

0.5

0.6

0.7

0.8

λ

Satis

fact

ion

degr

ees

m1

m2

m3

m4

m5

Fig. 9. The satisfaction degrees obtained by the generalized hybrid weighted distance measure with different parameter value.

H. Liao et al. / Information Sciences 271 (2014) 125–142 141

Figs. 7–9 also imply some interesting results. When using different distance measures, we can see that the satisfactiondegrees are increasing or decreasing as the parameter k changes. For example, if we use the generalized hybrid weighteddistance measures to calculate the distances (shown as Fig. 9), the satisfaction degrees of m2 and m5 are monotonicallyincreasing as the parameter changes, while the satisfaction degrees of m1; m3 and m4 are monotonically decreasing. Similarresults can be derived from Figs. 7 and 8 as well. Hence, from this point of view, the parameter k can be regarded as a decisionmaker’s risk attitude. The proposed distance measures thus give the decision maker more choices as the parameter regardingto the decision maker’s risk preference is provided.

For the convenience of application, it is necessary for us to compare all of these distance and similarity measures pro-posed in this paper. All the measures presented in Section 3 are the ones between two HFLTSs, while the measures intro-duced in Section 4 are mainly about the distance and similarity measures between two collections of HFLTSs. Thus, onlythe measures in Section 4 have the weighted forms due to that there are different sorts of HFLTSs in each set H

jSðj ¼ 1;2Þ,

and the HFLTSs in the set may have different importance degrees. As for the distance measures of HFLTSs, the Hamming dis-tance and the Euclidean distance are the special cases of the generalized distance measure with k ¼ 1 and k ¼ 2 respectively.The generalized Hausdorff distance measure is also a special case of the generalized distance measure in a sense that L!1.Analogously, the Hamming–Hausdorff distance and the Euclidean–Hausdorff distance are the especial cases of the Hammingdistance and the Euclidean distance with L!1 respectively. The distance measures between two collections of HFLTSsdeveloped in Section 4 also have these properties, i.e., the generalized distance measure and the generalized Hausdorff dis-tance measure are the basic types of distances between two collections of HFLTSs and the others are their special cases.When the weights are given in discrete forms, we can use those measures introduced in Section 4.1; while if the weightsare provided in continuous forms, then the continuous distance measures given in Section 4.2 can be employed.

6. Conclusions

In this paper, we have investigated different types of distance and similarity measures for HFLTSs. After giving the basicaxioms for distance and similarity measures, we have developed a family of distance and similarity measures for HFLTSsbased on the well known Hamming distance, the Euclidean distance, the Hausdorff distance and their generalizations. Sub-sequently, with respect to two collections of HFLTSs, we also have developed a variety of weighted distance and similaritymeasures for discrete cases and a series of continuous weighted distance and similarity measures for continuous cases. Itshould be pointed out that in this paper we have focused our attention on distance measures; while the corresponding sim-ilarity measures for HFLTSs can be obtained via the relationship between the distance measures and the similarity measures.It is also noted that in real applications, the lengths of two different HFLTSs are often different, and all the distance and sim-ilarity measures proposed in this paper are based on the assumption that the shorter one should be extended by adding theaverage value in it until both of them have the same length. Generally speaking, we can extend the shorter one by adding anyvalue in it until it has the same length of the longer one according to the decision maker’s preferences and actual situations,which is the practical meaning of the parameter n in Eq. (3). We have applied these proposed distance and similarity mea-sures to multi-criteria decision making. A practical example concerning the evaluation of the quality of movies has shownthe applicability and efficiency of the proposed approach. From the numerical results, we have seen that the parameter k canbe regarded as a decision maker’s risk attitude. As a result, our distance measures give decision makers more choices as theparameter regarding to decision makers’ risk preferences is provided.

In future, we may investigate the hybrid weighted distance and similarity measures between two collections of HFLTSs.Furthermore, we may apply our distance and similarity measures to other decision making methods, and the method todetermine the weights is also an issue to be investigated.

Acknowledgements

The authors would like to thank the editors and the anonymous referees for their insightful and constructive commentsand suggestions that have led to this improved version of the paper. The work was supported in part by the National NaturalScience Foundation of China (No. 61273209), the Excellent Ph.D. Thesis Foundation of Shanghai Jiao Tong University (No.20131216), and the Scholarship from China Scholarship Council (No. 201306230047).

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