Distance and similarity measures for hesitant fuzzy linguistic term sets ...

Information Sciences 271 (2014) 125–142

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Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making Huchang Liao a,c, Zeshui Xu b,⇑, Xiao-Jun Zeng c a

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

a r t i c l e

i n f o

Article history: Received 4 December 2012 Received in revised form 20 February 2014 Accepted 21 February 2014 Available online 1 March 2014 Keywords: Hesitant fuzzy linguistic term set Distance measure Similarity measure Multi-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’s hesitant preferences when assessing a linguistic variable, increase the flexibility of eliciting and representing linguistic information. The HFLTSs have attracted a lot of attention recently due to their distinguished power and efficiency in representing uncertainty and vagueness within the process of decision making. To enhance and extend the applicability of HFLTSs, this paper investigates and develops different types of distance and similarity measures for HFLTSs. The paper first proposes a family of distance and similarity measures between two HFLTSs. Then a variety of weighted or ordered weighted distance and similarity measures between two collections of HFLTSs are proposed and analyzed for discrete and continuous cases respectively. After that, the application of these measures to multi-criteria decision making problems is given. Based on the proposed distance and similarity measures, the satisfaction degrees for different alternatives are established and are then used to rank alternatives in multi-criteria decision making. Finally a practical example concerning the evaluation of the quality of movies is given to illustrate the applicability and advantage of the proposed approach and the differences between the proposed distance and similarity measures. Ó 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 a lot 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 membership degree 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 values between 0 and 1, it can express a decision maker’s hesitancy efficiently, especially when two or more sources of vagueness appear simultaneously. It should be noted that the HFS was introduced to handle the problems that are represented in quantitative situations. In many cases, however, uncertainty is produced by the vagueness of meanings whose nature is qualitative rather than quantitative [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. ⇑ Corresponding author. Tel.: +86 25 84483382. E-mail addresses: [email protected] (H. Liao), [email protected] (Z. Xu), [email protected] (X.-J. Zeng). http://dx.doi.org/10.1016/j.ins.2014.02.125 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.

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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], the proportional 2-tuple model [32], and so on [6]. However, all these extended models have some serious limitation due to the fact that they assess a linguistic variable by using a single linguistic term rather than following the information provided by decision makers regarding the linguistic variable. Because decision makers may consider several terms at the same time or need a complex linguistic term, such a single linguistic term is often insufficient or very hard to be determined. Motivated by 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 process. The HFLTS increases the flexibility and capability of eliciting and representing linguistic information. It permits decision makers to use several linguistic terms to assess a linguistic variable. Thus, it provides many advantages in depicting decision makers’ cognitions and preferences. Rodríguez et al. [26] applied HFLTSs to multi-criteria linguistic decision making problems in which decision makers can provide their assessments by linguistic expressions based on comparative terms, such as ‘‘between very low and medium’’, or by simple terms, such as ‘‘very low; low; medium; high; very high’’. By using HFLTSs and context-free grammar, Rodríguez et al. [27] then proposed a new linguistic group decision making model that facilitates the elicitation of flexible linguistic expressions close to human being’s cognitive models for expressing linguistic preferences. Liu and Rodríguez [19] presented a new representation of HFLTSs by means of a fuzzy envelope to carry out the computing with words process. Later, Zhu and Xu [53] introduced the hesitant fuzzy linguistic preference relation (HFLPR) as a tool to collect and represent decision makers’ preferences and then investigated the consistency of the HFLPR. In order to apply HFLTSs to solve multi-criteria decision making problems more effectively, we shall pay more attention to the basic characteristics of HFLTSs, in particular distance and similarity measures which are fundamentally important in many scientific fields, such as decision making, pattern recognition, 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 for HFLTSs, 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 term sets and hesitant fuzzy linguistic term sets. In Section 3, we first review some known distance and similarity measures and then give the definitions of distance and similarity measures for HFLTSs, based on which several distance and similarity measures for two HFLTSs are introduced. In Section 4, we focus on the distance and similarity measures for two collections of HFLTSs, 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 satisfaction degrees of different alternatives are defined in order to rank alternatives. A practical example concerning the evaluation of the quality of movies is then given to illustrate the applicability and advantage of the proposed approach and the different 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 knowledge by using linguistic terms that are close to human being’s cognitive processes. To model and manage such knowledge with uncertainty, the fuzzy linguistic approach which uses fuzzy set theory to model the linguistic information was proposed by Zadeh in [51]. The linguistic variable, defined as ‘‘a variable whose values are not numbers but words or sentences in a natural or artificial language’’, enhances the flexibility and applicability of the decision models and provides good application results in many 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 syntactic rule (which usually takes the form of a grammar) for generating the terms in TðHÞ; and M is a semantic rule for associating each 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 are different ways to choose the linguistic descriptors and to define their semantics [8,9,25,26,48]. The commonly used approaches 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 the linguistic term set; (2) semantics based on membership functions and a semantic rule; and (3) mixed semantics. We pay our attention 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 considering all 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 v ery low; s2 ¼ low; s3 ¼ medium; s4 ¼ high; s5 ¼ v ery high; s6 ¼ perfectg. Meanwhile, some other linguistic 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:

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S ¼ fsa ja 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 symmetrically around it. ss and ss are the lower and upper bounds of linguistic labels where s is a positive integer. S satisfies the following conditions: (1) If a > b, then sa > sb . (2) The negation operator is defined as: neg ðsa Þ ¼ sa , especially, neg ðs0 Þ ¼ s0 . For example, when

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

S ¼ fs3 ¼ none; s2 ¼ v ery low; s1 ¼ low; s0 ¼ medium; s1 ¼ high; s2 ¼ v ery 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-symmetric 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 elicitation. The decision makers can use the fuzzy linguistic approach to express their assessments over a linguistic variable. For example, considering a person’s age as a linguistic variable, the linguistic term set could be given as TðageÞ ¼ fyoung; not young; v ery young; old; not old; v ery old; middleaged; not middle  aged; etc:g. In such a case, the numerical variable age, whose values are the numbers 0; 1; 2; 3; . . . ; 100, constitutes what may be called the base variable for age. The linguistic value (such as young) can be interpreted as a label for a fuzzy restriction on the values of the base variable. Such fuzzy restriction is characterized by a compatibility function associated each age in the interval ½0; 100 with a real number in the interval ½0; 1, which represents the compatibility of that age with the label young. For instance, the compatibility degrees of the numerical ages 20, 25 and 35 with the linguistic value young might be 1, 0.9 and 0.1, respectively. The compatibility function is in fact the membership function in Zadeh’s fuzzy set theory [51]. The triangular membership function (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, an approximation process must be employed to translate the results into the initial expression domain, which consequently produces the loss of information. In order to preserve all given information, Xu [37] extended the discrete linguistic term set S to the continuous linguistic term set S ¼ fsa ja 2 ½q; qg, where qðq > sÞ is a sufficiently large positive integer. In general, the linguistic term sa ðsa 2 SÞ is given by the decision maker, while the extended linguistic term (also named virtual linguistic term) sa ðsa 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) (2) (3) (4)

sa  sb ¼ saþb ; ksa ¼ ska ; ðk1 þ k2 Þsa ¼ k1 sa  k2 sa ; kðsa  sb Þ ¼ ksa  ksb .

none

very low

low

medium

high

very high

s−3

s−2

s−1

s0

s1

s2

perfect s3

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

s0

s−3

s −2

s −1

none

very low

low

medium

high

very high

perfect

0

0.17

0.33

0.5

0.67

0.83

1

s1

s2

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

s3

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s−1.6

s1.3

s−3

s−2

s−1

s0

s1

s2

s3

none

very low

low

medium

high

very high

perfect

0 0.068 0.17

0.33 0.399 0.5 0.551 0.67

0.83 0.881

1

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 0:5 þ ð1:3  1Þ  0:17 and 0:881 ¼ 0:83 þ ð1:3  1Þ  0:17.

It is noted that the operational laws (1) and (2) hold only from a theoretical point of view, and we usually do not use them in practice because it makes no sense if we add the linguistic term ‘‘young’’ to the linguistic term ‘‘very young’’. When computing linguistic variables in practical applications, the fusing process is always associated with a weighting vector [41], that is, the fusion is in fact a convex combination. For this reason, we only need to use the operational laws (3) and (4). Since ss and ss are the lower and upper bounds of the linguistic labels determined by the decision maker, it is impossible that the initial value of a linguistic variable given by the decision maker is greater than ss or lower than ss . Hence, the final results are still within the interval ½ss ; ss . One question that has triggered off some discussions over the virtual linguistic term set concerns their corresponding fuzzy semantics representation and linguistic syntax [25]. In fact, we still can construct the mapping between virtual linguistic terms and their corresponding semantics. For example, if we obtain two virtual linguistic terms s1:3 and s1:6 , then the semantics of them can be represented by the compatibility functions with the triangular forms C s1:3 ¼ ð0:551; 0:721; 0:881Þ and C s1:6 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 psychometric scale commonly involved in the research based on survey questionnaires. The range of Likert scale only captures the discrete intensity of respondents’ feelings for a given item. Thus, the Likert scale cannot use a function to represent the intensity, while the (virtual) linguistic term set is a fuzzy linguistic approach, which utilizes the compatibility function to express the compatibility between the basic variable value and the linguistic term. For example, in Fig. 3, the compatibility functions of the linguistic terms are ones with the triangular type C sa ¼ ðai ; bi ; ci Þ. Furthermore, the (virtual) linguistic term set 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 possible values, 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 returns a subset of ½0; 1. To be easily understood, Xu and Xia [33,46] expressed the HFS by a mathematical symbol:

E ¼ fhx; hE ðxÞijx 2 Xg

ð2Þ

where hE ðxÞ 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 terms to assess a linguistic variable. Hence, motivated by the idea of HFSs, Rodríguez et al. [26] introduced the hesitant fuzzy linguistic term set (HFLTS), whose envelope is an uncertain linguistic variable [34]. Definition 3 [26]. Let S ¼ fs0 ; . . . ; ss g be a linguistic term set, a hesitant fuzzy linguistic term set (HFLTS), HS , is an ordered finite subset of the consecutive linguistic terms of S. For the linguistic term set S ¼ fs0 ; . . . ; ss g in Definition 3, as its subscripts are not symmetric, then some problems will arise. For example, if we take S ¼ fs0 ¼ none; s1 ¼ v ery low; s2 ¼ low; s3 ¼ medium; s4 ¼ high; s5 ¼ v ery 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 our intuition. To circumvent this issue, we can replace the linguistic term set S ¼ fs0 ; . . . ; ss g by the subscript-symmetric linguistic term set S ¼ fsa ja ¼ s; . . . ; 1; 0; 1; . . . ; sg. It is worth pointing out that using different types of linguistic term sets does not influence the theoretical derivation of distance and similarity measures for HFLTSs. The only difference takes place in their distinct semantics.

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Let S ¼ fsa ja ¼ s; . . . ; 1; 0; 1; . . . ; sg be a linguistic term set. The HFLTS HS for a linguistic variable v 2 V can then be represented mathematically as HS ðv Þ. For the convenience of statement, we call H ¼ fHS ðv Þjv 2 Vg a set of HFLTSs. The aim of introducing HFLTS is to improve the elicitation of linguistic information, mainly when decision makers hesitate between several 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 v ery low; s2 ¼ low; s3 ¼ medium; s4 ¼ high; s5 ¼ v ery high; s6 ¼ perfectg be a linguistic 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 v ery high. Then, according to n o the definition of HFLTS, the above linguistic information can be represented as HS ¼ H1S ; H2S ; H3S ; H4S with H1S ¼ fs4 g; H2S ¼ fs0 ; s1 ; s2 ; s3 g; H3S ¼ fs4 ; s5 ; s6 g and H4S ¼ fs3 ; s4 ; s5 g 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 H2S , the following operations are defined: (1) Lower bound: H S ¼ minðsi Þ ¼ sj ; si 2 hS and si P sj ; 8i. (2) Upper bound: Hþ S ¼ maxðsi Þ ¼ sj ; si 2 hS and si 6 sj ; 8i. (3) HcS ¼ S  HS ¼ fsi jsi 2 S and si R HS g. n o (4) H1S [ H2S ¼ si jsi 2 H1S or si 2 H2S . n o (5) H1S \ H2S ¼ si jsi 2 H1S and si 2 H2S . From Example 1, we can see that different HFLTSs have different numbers of linguistic terms in most cases. In order to operate correctly when comparing two HFLTSs, Zhu and Xu [53] introduced a method to add linguistic terms in a HFLTS. þ  Let b ¼ fbl jl ¼ 1; . . . ; #bg be a HFLTS (#b is the number of linguistic terms in b), b and b be the maximum and minimum linguistic terms in b respectively, and nð0 6 n 6 1Þ be an optimized parameter. Then we can add the linguistic term  

þ



b ¼ nb  ð1  nÞb

ð3Þ 

into the HFLTS. The max, min and the averaged linguistic terms correspond with b ; b , and bA respectively, where  þ  þ   b ¼ b ; b ¼ b , and bA ¼ 12 ðb  b Þ. It’s obvious that b and b correspond with the optimism and pessimism rules, respectively. The optimized parameter is used to reflect decision makers’ risk preferences. Without loss of generality, in this paper, we assume n ¼ 12. 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 different arguments. Up to now, many scholars have paid great attention to this issue and have achieved many results, which can be roughly classified into two sorts: one sort is based on the traditional distance measures, such as the Hamming distance, the Euclidean distance, and the Hausdorff metric [7,11,12,28,36,43]. The other is on the basis of some weighted distance operators, such as the ordered weighted distance measures [42], the hybrid weighted distance measures [39], and the fuzzy 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 Bustince [2] defined the normalized Hamming distance and the normalized Euclidean distance, which only involve the first two parameters, the membership degree and the nonmembership degree in describing an IFS. Considering the geometrical representation of distances between IFSs, Szmidt and Kacprzyk [28] showed that the third parameter (the degree of indeterminacy) should not be omitted when calculating distances between IFSs. Then they defined some similarity measures for IFSs based on the proposed distance measures [29]. Hung and Yang [11] presented a method to calculate the distances between IFSs on the basis of the Hausdorff distance and then defined the similarity measures for IFSs. Grzegorzewski [7] also proposed some distance measures based on the Hausdorff metric. Interval-valued intuitionistic fuzzy sets (IVIFSs) [1] are a generalized case of IFSs whose values are intervals rather than exact numbers. Thus their distance and similarity measures can be derived in analogy to IFSs. Xu and Chen [43] gave a comprehensive overview of distance and similarity measures of IFSs and defined some continuous distance and similarity measures for IFSs and IVIFSs. Xu [37] defined the concepts of deviation degree and similarity degree between two linguistic values, as well as the concepts of deviation degree and similarity degree between two linguistic preference relations. Xu and Xia [46] proposed a variety of distance measures for HFSs, based on which the corresponding similarity measures were obtained. On the other hand, based on the weighting operators, distinct forms of distance and similarity measures have been proposed. Based on the Choquet integral [5] with respect to the non-monotonic fuzzy measure, Narukawa and Torra [23] introduced a weighted distance measure for IFSs. Based on the geometric distance model, Xu [38] generalized Szmidt and

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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 linguistic 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] developed a 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 they give good inspiration for us to develop the distance and similarity measures for HFLTSs. Motivated by the above analysis, we can set out our investigation of the distance and similarity measures over HFLTSs from two aspects, i.e., the extensions of traditional 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 fsa ja ¼ s; . . . ; 1; 0; 1; . . . ; sg be a linguistic term set, H1S and H2S be two HFLTSs, then the distance   measure between H1S and H2S is defined as d H1S ; H2S , which satisfies:   (1) 0 6 d H1S ; H2S 6 1;   (2) d H1S ; H2S ¼ 0 if and only if H1S ¼ H2S ;     (3) d H1S ; H2S ¼ d H2S ; H1S .

Definition 6. Let S ¼ fsa ja ¼ s; . . . ; 1; 0; 1; . . . ; sg be a linguistic term set, H1S and H2S be two HFLTSs, then the similarity   measure between H1S and H2S is defined as q H1S ; H2S , which satisfies:   (1) 0 6 q H1S ; H2S 6 1;   (2) q H1S ; H2S ¼ 1 if and only if H1S ¼ H2S ;     (3) q H1S ; H2S ¼ q H2S ; 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 other word, each different form of distance or similarity measure should satisfy these three conditions respectively. Noticing that the relationship between distance and similarity measures is







q H1S ; H2S ¼ 1  d H1S ; H2S



ð4Þ

in this paper, we mainly discuss the distance measures of HFLTSs, and the corresponding similarity measures can be obtained 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 fsa ja ¼ s; . . . ; 1; 0; 1; . . . ; sg be a linguistic term set, sa ; sb 2 S be two linguistic terms, then the distance measure between sa and sb is

dðsa ; sb Þ ¼

ja  bj 2s þ 1

ð5Þ

where 2s þ 1 is the number of linguistic terms in the set S. n o Let S ¼ fsa ja ¼ s; . . . ; 1; 0; 1; . . . ; sg be a linguistic term set, H1S ðxi Þ ¼ [s 1 2H1 sd1 jl ¼ 1; . . . ; #H1S (#H1S be the number of S l d n o l linguistic terms in H1S ) and H2S ðxi Þ ¼ [s 2 2H2 sd2 jl ¼ 1; . . . ; #H2S be two HFLTSs on X ¼ fx1 ; x2 ; . . . ; xn g, where #H1S ¼ #H2S ¼ L d l

S

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 distance for HFLTSs. The Hamming distance of H1S ðxi Þ and H2S ðxi Þ can be defined as:

  L  1   1X dl  d2l  dhd H1S ðxi Þ; H2S ðxi Þ ¼ L l¼1 2s þ 1

ð6Þ

and the Euclidean distance of H1S ðxi Þ and H2S ðxi Þ can be defined as:

0  1 ! 11=2 L   d  d2  2 X 1 1 2 l l A ded HS ðxi Þ; HS ðxi Þ ¼ @ L l¼1 2s þ 1

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 d2l 6 s, then we have 0 6 d1l  d2l  6 2s. Thus,     0 6 dhd H1S ; H2S 6 1. In addition, if H1S ¼ H2S , i.e., d1l ¼ d2l , for l ¼ 1; . . . ; L, then dhd H1S ; H2S ¼ 0. On the other hand, if       dhd H1S ; H2S ¼ 0, then we can derive that H1S ¼ H2S . It is obvious that dhd H1S ; H2S ¼ dhd H2S ; H1S . Hence, the Hamming distance measure satisfies the axioms of distance measure for HFLTSs given as Definition 5. All the other distance measures can be verified 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 the following generalized distance measure:

0  1 ! 11=k L   d  d2  k X 1 1 2 l l A dgd HS ðxi Þ; HS ðxi Þ ¼ @ L l¼1 2s þ 1

ð8Þ

where k > 0. In particular, if k ¼ 1, then the above generalized distance becomes the Hamming distance; If k ¼ 2, then the above generalized distance becomes the Euclidean distance. Example 2. Let S ¼ fsa ja ¼ 3; . . . ; 1; 0; 1; . . . ; 3g be a linguistic term set, H1S ¼ fs1 ; s2 g and H2S ¼ fs3 ; s1 ; s3 g be two HFLTSs on S. Further we can extend H1S to H1S ¼ fs1 ; s1:5 ; s2 g by adding the linguistic term s1:5 . Thus, the generalized distance between H1S and H2S is

  ded H1S ; H2S ¼

 k  k  k !!1=k j1  ð3Þj j1:5  ð1Þj j2  3j þ þ 7 7 7

1 3

If k ¼ 1, then the Hamming distance between H1S and H2S is

  1 j1  ð3Þj j1:5  ð1Þj j2  3j ¼ 0:3571 dhd H1S ; H2S ¼ þ þ 3 7 7 7 If k ¼ 2, the Euclidean distance of H1S and H2S is

ded ðH1S ; H2S Þ

¼

1 3

 2  2  2 !!1=2 j1  ð3Þj j1:5  ð1Þj j2  3j þ þ ¼ 0:3977 7 7 7

Similarly, the Hausdorff distance measure can be introduced to HFLTS. For two HFLTSs H1S ðxi Þ and H2S ðxi Þ, the generalized Hausdorff distance measure can be defined as:



dghaud H1S ðxi Þ; H2S ðxi Þ



0 ¼ @ max

l¼1;2;...;L

 1 ! 11=k d  d2  k l l A 2s þ 1

ð9Þ

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

 1    d  d2  l dhhaud H1S ðxi Þ; H2S ðxi Þ ¼ max l l¼1;2;...;L 2s þ 1

ð10Þ

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



dehaud H1S ðxi Þ; H2S ðxi Þ



0 ¼ @ max

l¼1;2;...;L

 1 ! 11=2 d  d2  2 l l A 2s þ 1

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 ðxi Þ and H2S ðxi Þ:

   1 ! L  1   1 1X d  d2  dl  d2l  1 2 l l dhhd HS ðxi Þ; HS ðxi Þ ¼ þ max l¼1;2;...;L 2s þ 1 2 L l¼1 2s þ 1

ð12Þ

(2) The hybrid Euclidean distance between H1S ðxi Þ and H2S ðxi Þ:

0 0  1 !  1 ! 111=2 L   d  d2  2 d  d2  2 X 1 1 1 2 l l l l AA dhed HS ðxi Þ; HS ðxi Þ ¼ @ @ þ max l¼1;2;...;L 2 L l¼1 2s þ 1 2s þ 1

ð13Þ

132

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

(3) The generalized hybrid distance between H1S ðxi Þ and H2S ðxi Þ:

0 0  1 !  1 ! 111=k L   d  d2  k d  d2  k X 1 1 1 2 l l l l AA dghd HS ðxi Þ; HS ðxi Þ ¼ @ @ þ max l¼1;2;...;L 2s þ 1 2 L l¼1 2s þ 1

ð14Þ

where k > 0.

Example 3 (Continue with Example 2). According to Eq. (14), the generalized Hausdorff distance between H1S and H2S is

dghaud ðH1S ; H2S Þ

max

¼

(

j1  ð3Þj 7

k  k  k )!1=k j1:5  ð1Þj j2  3j ; ; 7 7

Then, if k ¼ 1, then the Hamming–Hausdorff distance between H1S and H2S is

    j1  ð3Þj j1:5  ð1Þj j2  3j ¼ 0:5714 dhhaud H1S ; H2S ¼ max ; ; 7 7 7

if k ¼ 2, then the Euclidean–Hausdorff distance between H1S and H2S is

  dehaud H1S ; H2S ¼

max

( 2  2  2 )!1=2 j1  ð3Þj j1:5  ð1Þj j2  3j ; ; ¼ 0:5714 7 7 7

The generalized hybrid distance between H1S and H2S is

  dghd H1S ; H2S ¼

1 1 2 3

(      )!!1=k  k  k  k ! k k k 4 2:5 1 4 2:5 1 þ max þ þ ; ; 7 7 7 7 7 7

If k ¼ 1, then the hybrid Hamming distance between H1S and H2S is

  1 dhhd H1S ; H2S ¼ ð0:3571 þ 0:5714Þ ¼ 0:4643 2 If k ¼ 2, then the hybrid Euclidean distance between H1S and H2S is

1=2   1 dhed H1S ; H2S ¼ ð0:1582 þ 0:3265Þ ¼ 0:4733 2 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 single linguistic variable. However, in many real applications such as multi-criteria decision making, the objects/alternatives are often 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 information of alternatives with respect to different criteria is represented by several collections of HFLTSs, we need to calculate the distance and similarity measures between these collections of HFLTSs in order to compare the considered alternatives. In this 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 n o and Let S ¼ fsa ja ¼ s; . . . ; 1; 0; 1; . . . ; sg be a linguistic term set. For two collections of HFLTSs H1S ¼ H11 ; H12 ; . . . ; H1m S S S n o Pm T 21 22 2m 2 HS ¼ HS ; HS ; . . . ; HS with the associated weighting vector w ¼ ðw1 ; w2 ; . . . ; wm Þ , where 0 6 wj 6 1 and j¼1 wj ¼ 1, a generalized weighted distance measure between H1S and H2S is defined as:

0 1k 11=k 0 1j 2j  m L d  d  X X

B l  wj @ l AC dgwd H1S ; H2S ¼ @ A L s þ 1 2 j¼1 l¼1

ð15Þ

and a generalized weighted Hausdorff distance measure between H1S and H2S is defined as:



2

dgwhaud H1S ; HS

0 1k 11=k 0 1j  m dl  d2j  X l  B AC ¼ @ wj max @ A l¼1;2;...;L s þ 1 2 j¼1

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 H1S and H2S :



2

dwhd H1S ; HS ¼

   1j 2j  m L d  d  X X l l wj L

j¼1

l¼1

2s þ 1

ð17Þ

and the weighted Hamming–Hausdorff distance between H1S and H2S :



   1j 2j  m dl  dl  X wj max ¼ l¼1;2;...;L 2s þ 1 j¼1

2

dwhhaud H1S ; HS

ð18Þ

If k ¼ 2, then we can get the weighted Euclidean distance between H1S and H2S :



2

dwed H1S ; HS

0 12 11=2 0 1j 2j  m L d  d  X X l  wj B @ l A C ¼@ A L s þ 1 2 j¼1 l¼1

ð19Þ

and the weighted Euclidean–Hausdorff distance:

0



2

dwehaud H1S ; HS

12 11=2 0 1j  m dl  d2j  X l  B A C ¼ @ wj max @ A l¼1;2;...;L 2s þ 1 j¼1

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 H2S :

 1   0  1j  1j 2j  2j  m L d  d  dl  dl  1 2 X l l wj @1 X A dhwhd HS ; HS ¼ þ max l¼1;2;...;L 2s þ 1 2 L l¼1 2s þ 1 j¼1

ð21Þ

(2) The hybrid weighted Euclidean distance between H1S and H2S :

0 0 12 12 111=2 0 1j 0 1j 2j  2j  m L dl  dl  dl  dl  CC 1 2 BX wj B1 X @ A þ max @ A AA dhwed HS ; HS ¼ @ @ l¼1;2;...;L 2 L l¼1 2s þ 1 2s þ 1 j¼1

ð22Þ

(3) The generalized hybrid weighted distance between H1S and H2S :

0 0 1k 1k 111=k 0 1j 0 1j 2j  2j  m L dl  dl  dl  dl  CC 1 2 BX wj B1 X @ A þ x @ A AA dghwd HS ; HS ¼ @ @ l¼1;2;...;L 2 L l¼1 2s þ 1 2s þ 1 j¼1

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 the universe of discourse and the weights of elements are continuous. This subsection focuses on this case. Rb Let x 2 ½a; b, and the weight of x be wðxÞ, where wðxÞ 2 ½0; 1 and a wðxÞdx ¼ 1. Let H1S and H2S be two collections of HFLTSs over the element x. Then, in analogy to the above analysis, we can introduce a continuous weighted Hamming distance measure, a continuous weighted Euclidean distance measure, and a generalized continuous weighted distance measure between two collections of HFLTSs H1S and H2S , which are shown as follows, respectively:



dcwhd H1S ; H2S ¼

Z

b

wðxÞ

a

0 Z 1 2

dcwed HS ; HS ¼ @

  L  1 dl ðxÞ  d2l ðxÞ 1X dx L l¼1 2s þ 1

 1 ! 11=2 L d ðxÞ  d2 ðxÞ 2 1X l l wðxÞ dxA L l¼1 2s þ 1 a 0  1 ! 11=k Z b L d ðxÞ  d2 ðxÞ k X 1 2

1 l l wðxÞ dxA dgcwd HS ; HS ¼ @ L l¼1 2s þ 1 a

ð24Þ

b

ð25Þ

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

 1  Z b X 2 1 1 L dl ðxÞ  dl ðxÞ dx b  a a L l¼1 2s þ 1 0  1 ! 11=2 Z b X L d ðxÞ  d2 ðxÞ 2 1 2

1 1 l l dxA dcned HS ; HS ¼ @ b  a a L l¼1 2s þ 1 0  1 ! 11=k Z b X L d ðxÞ  d2 ðxÞ k 1 2

1 1 l l dxA dgcnd HS ; HS ¼ @ b  a a L l¼1 2s þ 1



dcnhd H1S ; H2S ¼

ð27Þ

ð28Þ

ð29Þ

where k > 0. Now we consider the Hausdorff metric. Similar to the above, a generalized continuous weighted distance measure, a continuous weighted Hamming–Hausdorff distance measure and a continuous weighted Euclidean–Hausdorff distance measure between two collections of HFLTSs H1S and H2S can be obtained as follows:

0 Z 1 2

dgcwhaud HS ; HS ¼ @

 1 ! 11=k d ðxÞ  d2 ðxÞ k l l wðxÞ max dxA l¼1;2;...;L 2s þ 1 a  1  Z b d ðxÞ  d2 ðxÞ

l dcwhhaud H1S ; H2S ¼ wðxÞ max l dx l¼1;2;...;L 2s þ 1 a 0  1 ! 11=2 Z b d ðxÞ  d2 ðxÞ 2 1 2

l l wðxÞ max dxA dcwehaud HS ; HS ¼ @ l¼1;2;...;L 2s þ 1 a b

ð30Þ

ð31Þ

ð32Þ

If wðxÞ ¼ 1=ðb  aÞ; 8 x 2 ½a; b, then Eqs. (30)–(32) reduce to a generalized continuous normalized distance measure, a continuous normalized Hamming–Hausdorff distance measure and a continuous normalized Euclidean–Hausdorff distance measure between two collections of HFLTSs H1S and H2S respectively, which can be shown as follows:

0  1 ! 11=k Z b d ðxÞ  d2 ðxÞ k 1 2

1 l l dgcwhaud HS ; HS ¼ @ max dxA b  a a l¼1;2;...;L 2s þ 1  1  Z b d ðxÞ  d2 ðxÞ 1 2

1 l l max dx dcwhhaud HS ; HS ¼ b  a a l¼1;2;...;L 2s þ 1 0  1 ! 11=2 Z b d ðxÞ  d2 ðxÞ 2 1 2

1 l l dcwehaud HS ; HS ¼ @ max dxA b  a a l¼1;2;...;L 2s þ 1

ð33Þ

ð34Þ

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 continuous weighted distance between two collections of HFLTSs H1S and H2S , which are shown below:

   1 ! L  1 d ðxÞ  d2 ðxÞ dl ðxÞ  d2l ðxÞ wðxÞ 1 X l l dx þ max ¼ l¼1;2;...;L 2 L l¼1 2s þ 1 2s þ 1 a 0 0  1 !  1 ! 1 11=2 Z b L d ðxÞ  d2 ðxÞ 2 d ðxÞ  d2 ðxÞ 2 1 2

wðxÞ @1 X l l l l @ AdxA dhcwed HS ; HS ¼ þ max l¼1;2;...;L 2 L l¼1 2s þ 1 2s þ 1 a 0 0  1 !  1 ! 1 11=k Z b L d ðxÞ  d2 ðxÞ k d ðxÞ  d2 ðxÞ k X 1 2

wðxÞ 1 l l l l @ AdxA þ max dghcwd HS ; HS ¼ @ l¼1;2;...;L 2 L l¼1 2s þ 1 2s þ 1 a

dhcwhd H1S ; H2S



Z

b

ð36Þ

ð37Þ

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 measure, a hybrid continuous normalized Euclidean distance measure and a generalized hybrid continuous normalized distance between two collections of HFLTSs H1S and H2S respectively:

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

   1 ! Z b L  1 d ðxÞ  d2 ðxÞ dl ðxÞ  d2l ðxÞ 1 1X l dx þ max l l¼1;2;...;L 2ðb  aÞ a L l¼1 2s þ 1 2s þ 1 0 0  1 !  1 ! 1 11=2 Z b L d ðxÞ  d2 ðxÞ 2 d ðxÞ  d2 ðxÞ 2 X 1 2

1 1 l l l l @ AdxA dhcwed HS ; HS ¼ @ þ max l¼1;2;...;L 2ðb  aÞ a L l¼1 2s þ 1 2s þ 1 0 0  1 !  1 ! 1 11=k Z b L d ðxÞ  d2 ðxÞ k d ðxÞ  d2 ðxÞ k X 1 2

1 1 l l l l @ AdxA þ max dghcwd HS ; HS ¼ @ l¼1;2;...;L 2ðb  aÞ a L l¼1 2s þ 1 2s þ 1



dhcwhd H1S ; H2S ¼

135

ð39Þ

ð40Þ

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 by many scholars and have been applied to fuzzy sets [40], linguistic fuzzy sets [40,41,45] and HFSs [46]. The prominent characteristic of ordered weighted distance is that it can alleviate/intensify the influence of unduly large/small deviations on the aggregation results by assigning them low/high weights. This desired property makes the ordered weighted distance measure very useful in realistic decision making problems. In [40], Xu considered the situations with linguistic, interval or fuzzy preference information and developed some fuzzy ordered distance measures. Xu and Wang [45] developed the linguistic ordered weighted distance operator and studied its properties, such as commutativity, monotonicity, idempotency and boundedness. 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 of HFLTSs H1S and H2S :

0 0 1k 111=k 0 1rðjÞ 2rðjÞ  m L d  d   CC X 1 2 BX l l B1 @ A AA dgowd HS ; HS ¼ @ wj @ L s þ 1 2 j¼1 l¼1 where k > 0 and

ð42Þ

rðjÞ : ð1; 2; . . . ; mÞ ! ð1; 2; . . . ; mÞ is a permutation such that

1k 1k 0 1rðjÞ 0 1rðjþ1Þ 2rðjÞ  2rðjþ1Þ  L L  dl   dl dl dl  X 1X 1 @ A P @ A; L l¼1 L l¼1 2s þ 1 2s þ 1

j ¼ 1; 2; . . . ; m

ð43Þ

In particular, if k ¼ 1, then the generalized ordered weighted distance becomes the ordered weighted Hamming distance between H1S and H2S :

   1rðjÞ 2rðjÞ  m L d  dl  1 2 X l wj X dowhd HS ; HS ¼ L l¼1 2s þ 1 j¼1

ð44Þ

If k ¼ 2, then the generalized ordered weighted distance becomes the ordered weighted Euclidean distance between H1S and H2S :

0 0 12 111=2 0 1rðjÞ 2rðjÞ  m L d  d  CC  X 1 2 BX l B1 @ l A AA dowed HS ; HS ¼ @ wj @ 2s þ 1 L l¼1 j¼1

ð45Þ

Based on the Hausdorff metric, we can also develop a generalized ordered weighted Hausdorff distance between two collections of HFLTSs H1S and H2S :



2

dgowhaud H1S ; HS

0 1k 11=k 0 1r_ ðjÞ 2r_ ðjÞ  m dl  dl   X B AC ¼ @ wj max @ A l¼1;2;...;L 2s þ 1 j¼1

ð46Þ

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

1k 1k 0 1r_ ðjÞ 0 1r_ ðjþ1Þ 2r_ ðjÞ  2r_ ðjþ1Þ   dl   dl dl dl  A P max @ A; max @ l¼1;2;...;L l¼1;2;...;L 2s þ 1 2s þ 1

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–Hausdorff distance between H1S and H2S :

   1r_ ðjÞ 2r_ ðjÞ  m  dl  dl 1 2 X dowhhaud HS ; HS ¼ wj max l¼1;2;...;L 2s þ 1 j¼1

ð48Þ

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

0 12 11=2 0 1r_ ðjÞ 2r_ ðjÞ  m d  d   C X

B l l A A dowehaud H1S ; H2S ¼ @ wj max @ l¼1;2;...;L 2s þ 1 j¼1

ð49Þ

Certainly, we can also derive some hybrid ordered weighted distance measures via combining the above distance measures, such as: (1) The hybrid ordered weighted Hamming distance between H1S and H2S :



2

dhwhd H1S ; HS ¼

   1 € ðjÞ  € ðjÞ   1r€ ðjÞ  1r€ ðjÞ 2r 2r L d  dl   dl  dl X l 1 @ A þ max l¼1;2;...;L 2 L l¼1 2s þ 1 2s þ 1

m X wj j¼1

0

ð50Þ

(2) The hybrid ordered weighted Euclidean distance between H1S and H2S :

0



2

dhwed H1S ; HS

0 1 1 111=2 0 1r€ ðjÞ 0 1r€ ðjÞ € ðjÞ  2 € ðjÞ  2 2r 2r m L d  d d  d     CC X X l l wj B1 B @ l A þ max @ l A AA ¼@ @ l¼1;2;...;L 2 L l¼1 2s þ 1 2s þ 1 j¼1

ð51Þ

(3) The generalized hybrid ordered weighted distance between H1S and H2S :

0



2

dghwd H1S ; HS

0 1 1 111=k 0 1r€ ðjÞ 0 1r€ ðjÞ € ðjÞ  k € ðjÞ  k 2r 2r m L d  dl   dl  dl BX wj B1 X@ l C A þ max @ AC ¼@ @ AA l¼1;2;...;L L 2 s þ 1 s þ 1 2 2 j¼1 l¼1

ð52Þ

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

1 1 1 1 0 1r€ ðjÞ 0 1r€ ðjÞ 0 1r€ ðjþ1Þ 0 1r€ ðjþ1Þ € ðjÞ  2 € ðjÞ  2 € ðjþ1Þ  2 € ðjþ1Þ  2 2r 2r 2r 2r L L  dl   dl   dl  dl dl dl dl  dl  X 1X 1 @ A þ max @ A P @ A þ max @ A ; l¼1;2;...;L l¼1;2;...;L L l¼1 L l¼1 2s þ 1 2s þ 1 2s þ 1 2s þ 1 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 good alternative(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 problem with HFLTS information can be interpreted as follows: Suppose that a decision maker is asked to evaluate a set of alternatives X ¼ fx1 ; x2 ; . . . ; xn g with respect to several criteria cj ðj ¼ 1; 2; . . . ; mÞ. The criteria have a weighting vector P w ¼ ðw1 ; w2 ; . . . ; wm ÞT , where 0 6 wj 6 1 and m j¼1 wj ¼ 1. The decision maker might feel much easier and are more willing to give their assessments by providing some linguistic expressions or sentences. Such linguistic information can be transformed into HFLTSs by using the context-free grammar [26]. Hence, a judgment matrix with HFLTS information will be obtained as follows:

2

H11 S 6 21 6 HS 6 HS ¼ 6 . 6 . 4 .

Hn1 S

S

H12 S H22 S .. . Hn2 S

   H1m S

3

7 7    H2m S 7 .. .. 7 7 . . 5

ð54Þ

   Hnm S

where HijS ¼ s 2Hij fsdij jl ¼ 1; . . . ; #HijS g ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . mÞ is a HFLTS, denoting the degree that the alternative xi ij S l d l 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 ¼ minl¼1;...;#Hij fsdij g and the upper bound S l þ Hijþ S ¼ maxl¼1;...;#Hij fsdij g. Then, we can define the notions of the hesitant fuzzy linguistic positive ideal solution x and hesitant S l  fuzzy linguistic negative ideal solution x as follows, respectively:

n o 2þ mþ xþ ¼ H1þ S ; HS ; . . . ; H S

ð55Þ

n o 2 m x ¼ H1 S ; HS ; . . . ; H S

ð56Þ

and

where

Hjþ S ¼

8 max Hijþ max fsdij g for benefit criterion cj > S ¼ i¼1;2;...;n > > i¼1;2;...;n l > ij < l¼1;...;#H S

> min Hij min fsdij g for cos t criterion cj > S ¼ i¼1;2;...;n > > l : i¼1;2;...;n ij l¼1;...;#H

;

for j ¼ 1; 2; . . . ; m

ð57Þ

;

for j ¼ 1; 2; . . . ; m

ð58Þ

S

and

Hj S ¼

8 min Hij min fsdij g for benefit criterion cj > S ¼ i¼1;2;...;n > > i¼1;2;...;n l > ij < l¼1;...;#H S

> max Hijþ max fsdij g for cos t criterion cj > S ¼ i¼1;2;...;n > > l : i¼1;2;...;n ij l¼1;...;#H

S

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 fuzzy linguistic positive ideal solution xþ , and the distance between each alternative xi and the hesitant fuzzy linguistic negative ideal solution x , respectively. Intuitively, the smaller the distance dðxi ; xþ Þ, the better the alternative; while the larger the distance dðxi ; x Þ, the better the alternative. All the proposed distance measures in Section 4 can be used to define and calculate these distances. Motivated by the well known TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method [3,13], we take both distance dðxi ; xþ Þ 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 cj ðj ¼ 1; 2; . . . ; nÞ is defined as:

gðxi Þ ¼

ð1  hÞdðxi ; x Þ hdðxi ; xþ Þ þ ð1  hÞdðxi ; x Þ

ð59Þ

where the parameter h denotes the risk preferences of the decision maker: h > 0:5 means that the decision maker is pessimists; 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 gðxi Þ 6 1, for any h 2 ½0; 1; i ¼ 1; 2; . . . ; m. The higher the satisfaction degree, the better the alternative. Different distance measures proposed in the previous sections can be employed to define and calculate the satisfaction degrees. In the following, we consider a multi-criteria decision making problem that concerns the evaluation of the 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 movies m1 ; m2 ; . . . ; m5 with respect to four criteria: story ðf1 Þ, acting (f2 Þ, visuals ðf3 Þ and direction ðf4 Þ. The weighing vector of these four 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 of the users who give the ratings. Since these criteria are all qualitative, it is convenient and only feasible for the decision makers to express their feelings by using linguistic terms. As pointed out by Miller [22], most decision makers cannot handle more than 9 factors when making their decision. Hence, the company constructs a seven point linguistic scale to assess the movies, which is S ¼ fs3 ¼ terrible; s2 ¼ v ery bad; s1 ¼ bad; s0 ¼ medium; s1 ¼ well; s2 ¼ v ery well; s3 ¼ perfectg. To get more objective and reasonable evaluation results, the company sets up a decision organization, which contains a group of decision makers, to assess the movies. In the process of evaluation, the decision makers may think several linguistic terms at the same time for a movie over a criterion. For example, the decision makers may consider that the acting of the movie m2 is between medium and perfect. Such a linguistic expression is common and more similar to human being’s cognition than just using a single linguistic term. The linguistic expression presented above is appropriate to be represented as a HFLTS fs0 ; s1 ; s2 ; s3 g. In addition, the decision makers in the decision organization sometimes may have different opinions on the movies, and sometimes they cannot reach some consensus results. For example, one decision maker may think the direction of the movie m2 is perfect (denoted as s3 Þ, and another person may think it is between medium and very

138

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

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

m1 m2 m3 m4 m5

f1

f2

f3

f4

fs2 ; s1 ; s0 g fs0 ; s1 ; s2 g fs2 ; s3 g fs0 ; s1 ; s2 g fs1 ; s0 g

fs0 ; s1 g fs1 ; s2 g fs1 ; s2 ; s3 g fs1 ; s0 ; s1 g fs0 ; s1 ; s2 g

fs0 ; s1 ; s2 g fs0 ; s1 g fs1 ; s2 g fs1 ; s2 ; s3 g fs0 ; s1 ; s2 g

fs1 ; s2 g fs0 ; s1 ; s2 g fs2 g fs1 ; s2 g fs0 ; s1 g

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

k¼1 k¼2 k¼4 k¼6 k ¼ 10

m1

m2

m3

m4

m5

Rankings

0.3158 0.3204 0.3177 0.3136 0.3062

0.5263 0.5420 0.5509 0.5549 0.5593

0.8158 0.7689 0.7413 0.7298 0.7207

0.5526 0.5357 0.5250 0.5206 0.5154

0.3421 0.3618 0.3778 0.3882 0.4017

m3 m3 m3 m3 m3

 m4  m2  m2  m2  m2

 m2  m4  m4  m4  m4

 m5  m5  m5  m5  m5

 m1  m1  m1  m1  m1

 m2  m4  m4  m4  m4

 m5  m5  m5  m5  m5

 m1  m1  m1  m1  m1

 m2  m4  m4  m4  m4

 m5  m5  m5  m5  m5

 m1  m1  m1  m1  m1

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

k¼1 k¼2 k¼4 k¼6 k ¼ 10

m1

m2

m3

m4

m5

Rankings

0.3704 0.3500 0.3279 0.3162 0.3046

0.5185 0.5341 0.5496 0.5563 0.5619

0.7500 0.7312 0.7217 0.7181 0.7153

0.5357 0.5285 0.5226 0.5198 0.5154

0.3846 0.3896 0.3972 0.4031 0.4116

m3 m3 m3 m3 m3

 m4  m2  m2  m2  m2

Table 4 The satisfaction degrees obtained by the generalized hybrid weighted distance measure. m1

m2

m3

m4

m5

Rankings

0.3516 0.3395 0.3246 0.3154 0.3051

0.5165 0.5362 0.5500 0.5559 0.5612

0.7976 0.7526 0.7298 0.7222 0.7171

0.5484 0.5321 0.5234 0.5201 0.5154

0.3596 0.3773 0.3905 0.3984 0.4086

m3 m3 m3 m3 m3

λ=1 λ=2 λ=4 λ=6 λ=10

0.8

0.7

Satisfaction degrees

k¼1 k¼2 k¼4 k¼6 k ¼ 10

0.6

0.5

0.4

0.3

0.2 1

2

3

4

5

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

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

 m4  m2  m2  m2  m2

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

λ=1 λ=2 λ=4 λ=6 λ=10

0.8

Satisfaction degrees

0.7

0.6

0.5

0.4

0.3

0.2

1

2

3

4

5

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

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

λ=1 λ=2 λ=4 λ=6 λ=10

0.8

Satisfaction degrees

0.7

0.6

0.5

0.4

0.3

0.2 1

2

3

4

5

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

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

m

0.8

m m

0.7 Satisfaction degrees

m m 0.6

1 2 3 4 5

0.5

0.4

0.3

0.2

1

2

4

6

10

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

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

well ðfs0 ; s1 ; s2 gÞ. If they cannot persuade each other, then we can represent the assessment as a HFLTS fs0 ; s1 ; s2 ; s3 g. After discussion, the final assessments of these five movies can be established and a hesitant fuzzy linguistic judgment matrix can be 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 and the hesitant fuzzy linguistic negative ideal solution m , which can be conducted easily via (55)–(58) and shown as mþ ¼ ffs3 g; fs3 g; fs3 g; fs2 gg and m ¼ ffs2 g; fs1 g; fs0 g; fs0 gg. 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 gðmi Þ for each movie mi can be calculated using Eq. (59). Without loss of generality, we choose h ¼ 0:5. If we use the generalized weighted distance measure, the generalized weighted Hausdorff distance measure, and the generalized 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 parameter k 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 third movie is the best choice for the company. Such a conclusion can be drawn directly from Figs. 4–9, in which m3 is always at the top of the figures.

m

0.8

m m

0.7 Satisfaction degrees

m m 0.6

1 2 3 4 5

0.5

0.4

0.3

0.2

1

2

4

6

10

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

m

0.8

m m

0.7 Satisfaction degrees

m m 0.6

1 2 3 4 5

0.5

0.4

0.3

0.2 1

2

4

6

10

λ 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 satisfaction degrees are increasing or decreasing as the parameter k changes. For example, if we use the generalized hybrid weighted distance measures to calculate the distances (shown as Fig. 9), the satisfaction degrees of m2 and m5 are monotonically increasing as the parameter changes, while the satisfaction degrees of m1 ; m3 and m4 are monotonically decreasing. Similar results can be derived from Figs. 7 and 8 as well. Hence, from this point of view, the parameter k can be regarded as a decision maker’s risk attitude. The proposed distance measures thus give the decision maker more choices as the parameter regarding to 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 proposed in this paper. All the measures presented in Section 3 are the ones between two HFLTSs, while the measures introduced in Section 4 are mainly about the distance and similarity measures between two collections of HFLTSs. Thus, only the measures in Section 4 have the weighted forms due to that there are different sorts of HFLTSs in each set HjS ðj ¼ 1; 2Þ, and the HFLTSs in the set may have different importance degrees. As for the distance measures of HFLTSs, the Hamming distance 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 Hamming distance and the Euclidean distance with L ! 1 respectively. The distance measures between two collections of HFLTSs developed in Section 4 also have these properties, i.e., the generalized distance measure and the generalized Hausdorff distance 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 weights are 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 basic axioms for distance and similarity measures, we have developed a family of distance and similarity measures for HFLTSs based on the well known Hamming distance, the Euclidean distance, the Hausdorff distance and their generalizations. Subsequently, with respect to two collections of HFLTSs, we also have developed a variety of weighted distance and similarity measures for discrete cases and a series of continuous weighted distance and similarity measures for continuous cases. It should be pointed out that in this paper we have focused our attention on distance measures; while the corresponding similarity 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 similarity measures proposed in this paper are based on the assumption that the shorter one should be extended by adding the average value in it until both of them have the same length. Generally speaking, we can extend the shorter one by adding any value 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 measures to multi-criteria decision making. A practical example concerning the evaluation of the quality of movies has shown the applicability and efficiency of the proposed approach. From the numerical results, we have seen that the parameter k can be regarded as a decision maker’s risk attitude. As a result, our distance measures give decision makers more choices as the parameter 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 to determine 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 comments and suggestions that have led to this improved version of the paper. The work was supported in part by the National Natural Science 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). References [1] [2] [3] [4]

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