Research ArticlePairwise Comparison and Distance Measure ofHesitant Fuzzy Linguistic Term Sets

Han-Chen Huang1 and Xiaojun Yang2

1 Department of Tourism and MICE Chung Hua University Hsinchu 30012 Taiwan2 Luoyang Electronic Equipment Test Center Luoyang Henan 471003 China

Correspondence should be addressed to Xiaojun Yang yangxiaojun2007gmailcom

Received 10 April 2014 Accepted 3 July 2014 Published 4 August 2014

Academic Editor Ker-Wei Yu

Copyright copy 2014 H-C Huang and X Yang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

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

1 Introduction

Since Zadeh introduced fuzzy sets [1] in 1965 several exten-sions of this concept have been developed such as type-2fuzzy sets [2 3] and interval type-2 fuzzy sets [4] type-119899fuzzy sets [5] intuitionistic fuzzy sets [6 7] and interval-valued intuitionistic fuzzy sets [8] vague sets [9] (vaguesets are intuitionistic fuzzy sets [10]) fuzzy multisets [1112] nonstationary fuzzy sets [13] Cloud models [14ndash18](Cloud models are similar to nonstationary fuzzy sets andtype-2 fuzzy sets) and hesitant fuzzy sets [19 20] In thereal world there are many situations in which problemsmust deal with qualitative aspects represented by vagueand imprecise information So in these situations often theexperts are more accustomed to express their assessmentsusing linguistic terms rather than numerical values In [21ndash23] Zadeh introduced the concept of linguistic variableas ldquoa variable whose values are not numbers but wordsor sentences in a natural or artificial languagerdquo Linguisticvariable provides a means of approximate characterizationof phenomena which are too complex or too ill definedto be amenable to description in conventional quantitative

ways Since then fuzzy sets and linguistic variables havebeen widely used in describing linguistic information as theycan efficiently represent peoplersquos qualitative cognition of anobject or a concept [24] Thus linguistic approaches havebeen so far used successfully in a wide range of applicationssuch as information retrieval [25ndash28] data mining [29]clinical diagnosis [30 31] and subjective evaluation [32ndash37] especially in decision-making [38ndash49] Usually linguisticterms (words) are represented by fuzzy sets [50] type-2 fuzzysets [51] interval type-2 fuzzy sets [52ndash54] 2-tuple linguisticmodel [40 55] and so forth In these linguistic modelsan expert generally provides a single linguistic term as anexpression of hisher knowledge However just as Rodriguezet al [56] pointed out the expert may think of several termsat the same time or look for a more complex linguisticterm that is not defined in the linguistic term set to expresshisher opinion In order to cope with this situation theyrecently introduced the concept of hesitant fuzzy linguisticterm sets (HFLTSs) [56] under the idea of hesitant fuzzy setsintroduced in [19 20]

Similarly to a hesitant fuzzy set which permits themembership having a set of possible values an HFLTS allows

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 954040 8 pageshttpdxdoiorg1011552014954040

2 Mathematical Problems in Engineering

an expert hesitating among several values for a linguistic vari-able For example when people assess a qualitative criterionthey prefer to use a linguistic one such as ldquobetween mediumand very highrdquo which contains several linguistic terms119898119890119889119894119906119898 ℎ119894119892ℎ V119890119903119910ℎ119894119892ℎ rather than a single linguisticterm In practical decision-making process uncertainty andhesitancy are usually unavoidable problemsTheHFLTSs candeal with such uncertainty and hesitancy more objectivelyand thus it is very necessary to develop some theories aboutHFLTSs

Comparisons and distance measures used for measuringthe deviations of different arguments are fundamentallyimportant in a variety of applications In the existing litera-ture there are a number of studies on distance measures forintuitionistic fuzzy sets [57ndash60] interval-valued intuitionisticfuzzy sets [61] hesitant fuzzy sets [62 63] linguistic values[64 65] and so forth Nevertheless an HFLTS is a linguisticterm subset and the comparison among these elements isnot simple In [56] Rodriguez et al introduced the conceptof envelope for an HFLTS and then ranked HFLTSs usingthe preference degree method of interval values [66] Butbecause an HFLTS is a set of discrete linguistic terms itmay seem problematical using the preference degree methodfor continuous interval to compare these discrete terms ofHFLTSs Up to now just a few research has been done onthe distance measure of HFLTSs [67] Consequently it isvery necessary to develop some comparison methods anddistance measure methods for HFLTSs In [67] to calculatethe distance of two HFLTSs Liao et al extend the shorterHFLTS by adding any value in it until it has the same length ofthe longer one according to the decision-makerrsquos preferencesand actual situations In this paper we present a new com-parison method of HFLTSs based on pairwise comparisonsof each linguistic term in the two HFLTSs Then a distancemeasure method based on the pairwise comparison matrixof HFLTSs is proposed without adding any value Finallywe utilize the comparison method and distance measuremethod to develop some approaches to solve themulticriteriadecision-making problems under hesitant fuzzy linguisticenvironments

The rest of the paper is organized as follows In Section 2the concepts of hesitant fuzzy sets and HFLTSs are intro-duced also the defects of the previous comparison methodfor HFLTSs are analyzed according to an example Section 3describes the comparison and distance measure of HFLTSsbased on the proposed pairwise comparison method InSection 4 a multicriteria decision-making problem is shownto illustrate the detailed processes and effectiveness of tworanking methods which are based on the comparisons anddistance measures of HFLTSs respectively Finally Section 5draws our conclusions and presents suggestions for futureresearch

2 Preliminaries

21 Hesitant Fuzzy Sets Hesitant fuzzy sets (HFSs) were firstintroduced by Torra [19] and Torra and Narukawa [20] Themotivation is that when determining the membership degree

of an element into a set the difficulty is not because we havea margin of error (such as an interval) but because we haveseveral possible values

Definition 1 (see [19]) Let 119883 be a fixed set a hesitant fuzzyset (HFS) on 119883 is in terms of a function ℎ that when appliedto 119883 returns a subset of [0 1]

To be easily understood Zhu et al [68] represented theHFS as the following mathematical symbol

119864 = ⟨119909 ℎ (119909)⟩ | 119909 isin 119883 (1)

where ℎ(119909) is a set of some values in [0 1] denoting thepossible membership degrees of the element 119909 isin 119883 to the set119864 Liao et al [67] called ℎ(119909) a hesitant fuzzy element (HFE)

Example 2 Let ℎ = 02 03 04 then ℎ is an HFE

Definition 3 (see [69]) For an HFE ℎ the score function of ℎis defined as

119904 (ℎ) =

1

ℎsum

120574isinℎ

120574 (2)

where ℎ is the number of the elements in ℎ

For two HFEs ℎ1 and ℎ2 if 119904(ℎ1) gt 119904(ℎ2) then ℎ1 issuperior to ℎ2 denoted by ℎ1 ≻ ℎ2 if 119904(ℎ1) = 119904(ℎ2) thenℎ1 is indifferent with ℎ2 denoted by ℎ1 sim ℎ2

Example 4 Assume that we have three HFEs ℎ1 =

02 03 04 ℎ2 = 02 035 05 and ℎ3 = 03 04 thenaccording to the score function of HFE (2) and Definition 3we have 119904(ℎ1) = (02 + 03 + 04)3 = 03 119904(ℎ2) = (02 +

035 + 05)3 = 035 and 119904(ℎ3) = (03 + 04)2 = 035 Thus119904(ℎ2) = 119904(ℎ3) gt 119904(ℎ1) that is the ranking is ℎ2 sim ℎ3 ≻ ℎ1

The concept of HFS is very useful to express peoplersquoshesitancy in daily life So since it was introduced more andmore decision-making theories and methods under hesitantfuzzy environment have been developed [56 62 63 67ndash73]

22 Hesitant Fuzzy Linguistic Term Sets Similarly to theHFSan expertmay hesitate among several linguistic terms such asldquobetween medium and very highrdquo or ldquolower than mediumrdquoto assess a qualitative linguistic variable To deal with suchsituations Rodriguez et al [56] introduced the concept ofhesitant fuzzy linguistic term sets (HFLTSs)

Definition 5 (see [56]) Suppose that 119878 = 1199040 119904119892 is afinite and totally ordered discrete linguistic term set where119904119894 represents a possible value for a linguistic variable AnHFLTS 119867119878 is defined as an ordered finite subset of theconsecutive linguistic terms of 119878

It is required that the linguistic term set 119878 should satisfythe following characteristics

(1) the set is ordered 119904119894 gt 119904119895 if and only if 119894 gt 119895(2) there is a negation operator Neg(119904119894) = 119904119892minus119894

Mathematical Problems in Engineering 3

Example 6 Let 119878 be a linguistic term set 119878 = 1199040

n (nothing) 1199041 vl (very low) 1199042 l (low) 1199043 m (medium)1199044 h (high) 1199045 vh (very high) and 1199046 p (perfect) twodifferent HFLTSs might be 119867

1

119878= 1199041 vl 1199042 l 1199043 m and

1198672

119878= 1199043 m 1199044 h

Definition 7 One defines the number of linguistic terms inthe HFLTS 119867119878 as the cardinality of 119867119878 denoted by |119867119878| InExample 6 |1198671

119878| = 3 and |119867

2

119878| = 2

Definition 8 (see [56]) The lower bound 119867119878minus and upperbound 119867119878+ of the HFLTS 119867119878 are defined as 119867119878minus = min119904119894 |119904119894 isin 119867119878 and 119867119878+ = max119904119894 | 119904119894 isin 119867119878

Definition 9 (see [56]) The envelope of the HFLTS 119867119878env(119867119878) is defined as the linguistic interval [119867119878minus 119867119904+] =

[Ind(119867119878minus) Ind(119867119878+)] where Ind provides the index of thelinguistic term that is Ind(119904119894) = 119894 In Example 6 env(1198671

119878) =

[1199041 1199043] = [1 3] and env(1198672119878) = [1199043 1199044] = [3 4]

Based on the definition of envelope Rodriguez et al [56]compare twoHFLTSs using the comparisonmethod betweentwo numerical intervals introduced by Wang et al [66]

Definition 10 (see [66]) Letting 119860 = [1198861 1198862] and 119861 = [1198871 1198872]

be two intervals the preference degree of119860 over 119861 (or119860 gt 119861)is defined as

119875 (119860 gt 119861) =

max (0 1198862 minus 1198871) minus max (0 1198861 minus 1198872)

(1198862 minus 1198861) + (1198872 minus 1198871)

(3)

and the preference degree of 119861 over 119860 (or 119861 gt 119860) is definedas

119875 (119861 gt 119860) =

max (0 1198872 minus 1198861) minus max (0 1198871 minus 1198862)

(1198862 minus 1198861) + (1198872 minus 1198871)

(4)

Example 11 Let 1198671

119878= 1199041 1199042 1199043 119867

2

119878= 1199043 1199044 and

1198673

119878= 1199045 1199046 be three different HFLTSs on 119878 According to

Definition 9 we have env(1198672119878) = [1 3] env(1198672

119878) = [3 4]

and env(1198673119878) = [5 6] The preference degrees calculated by

Definition 10 (3) and (4) are

119875 (1198671

119878gt 1198672

119878) = 0 119875 (119867

2

119878gt 1198671

119878) = 1

119875 (1198671

119878gt 1198673

119878) = 0 119875 (119867

3

119878gt 1198671

119878) = 1

119875 (1198672

119878gt 1198673

119878) = 0 119875 (119867

3

119878gt 1198672

119878) = 1

(5)

From Example 11 mentioned above it can be observedthat when we compare two HFLTSs using the preferencedegree method there exist two defects as follows

(1) The result 119875(1198672

119878gt 119867

1

119878) = 1 indicates that 119867

2

119878

is absolutely superior to 1198671

119878 In fact both 119867

1

119878and 119867

2

119878

contain the linguistic term 1199043 It means that the value of alinguistic variable may be equal in these two cases Thus itis unreasonable to say that 1198672

119878is absolutely superior to 119867

1

119878

(2) The result 119875(1198673

119878gt 1198671

119878) = 119875(119867

2

119878gt 1198671

119878) = 1 meaning

that when comparedwith1198671

119878 the twoHFLTSs1198672

119878and119867

3

119878are

identical In fact 1198673119878is more superior to 119867

1

119878compared to 119867

2

119878

to 1198671

119878 Thus using the preference degree method to compare

HFLTSs may result in losing some important informationBased on the analysis mentioned above we think that it

is not suitable to compare discrete linguistic terms in HFLTSsusing the comparison method for continuous numericalintervals By the definition of an HFLTS we know thatevery linguistic term in it is a possible value of the linguisticinformation And noting that the twoHFLTSs for comparingmay have different lengths So when comparing twoHFLTSsit needs pairwise comparisons of each linguistic term in them

3 Comparison and DistanceMeasure of HFLTSs

31 Distance between Two Single Linguistic Terms Let 119904119894 119904119895 isin119878 be two linguistic terms Xu [64] defined the deviationmeasure between 119904119894 and 119904119895 as follows

119889 (119904119894 119904119895) =

1003816100381610038161003816119894 minus 119895

1003816100381610038161003816

119879

(6)

where 119879 is the cardinality of 119878 that is 119879 = |119878|If only one preestablished linguistic term set 119878 is used in

a decision-making model we can simply consider [49 65]119889(119904119894 119904119895) = |119894 minus 119895| = 119904|119894minus119895|

Definition 12 Letting 119904119894 119904119895 isin 119878 be two single linguistic termsthen we call

119889 (119904119894 119904119895) = 119904119894 minus 119904119895 = 119894 minus 119895 (7)

the distance between 119904119894 and 119904119895

The distance measure between 119904119894 and 119904119895 has a definitephysical implication and reflects the relative position anddistance between 119904119894 and 119904119895 If 119889(119904119894 119904119895) = 0 then 119904119894 = 119904119895 If119889(119904119894 119904119895) gt 0 then 119904119894 gt 119904119895 If 119889(119904119894 119904119895) lt 0 then 119904119894 lt 119904119895

Theorem13 Letting 119904119894 119904119895 119904119896 isin 119878 be three linguistic terms then(1) 119889(119904119894 119904119895) = minus119889(119904119895 119904119894)(2) (|119878| minus 1) le 119889(119904119894 119904119895) le (|119878| minus 1)(3) 119889(119904119894 119904119896) = 119889(119904119894 119904119895) + 119889(119904119895 119904119896)

Proof They are straightforward and thus omitted

32 Comparison of HFLTSs The comparison of HFLTSs isnecessary in many problems such as ranking and selectionHowever anHFLTS is a linguistic term subset which containsseveral linguistic terms and the comparison among HFLTSsis not simple Here a new comparison method of HFLTSswhich is based on pairwise comparisons of each linguisticterm in the two HFLTSs is put forward

Definition 14 Letting 1198671

119878and 119867

2

119878be two HFLTSs on 119878 then

one defines the pairwise comparison matrix between1198671

119878and

1198672

119878as follows

119862 (1198671

119878 1198672

119878) = [119889 (119904119894 119904119895)]|1198671

119878|times|1198672119878| 119904119894 isin 119867

1

119878 119904119895 isin 119867

2

119878 (8)

4 Mathematical Problems in Engineering

Remark 15 The number of linguistic terms in the twoHFLTSs 1198671

119878and 119867

2

119878 may be unequal that is |1198671

119878| = |119867

2

119878|

To deal with such situations usually it is necessary to extendthe shorter one by adding the stated value several times init [62 63] while our pairwise comparison method does notrequire this step

Remark 16 From Definition 14 we have [119862(1198671

119878 1198672

119878)] =

minus[119862(1198672

119878 1198671

119878)]

119879 where 119879 is the transpose operator of matrix

Example 17 Let 1198671119878

= 1199041 1199042 1199043 and 1198672

119878= 1199042 1199043 1199044 1199045 be

twoHFLTSs on 119878 According toDefinition 14 the comparisonmatrix 119862 between 119867

1

119878and 119867

2

119878is

1199041

1199042

1199043

1199042 1199043 1199044 1199045

[

[

minus1

0

1

minus2

minus1

0

minus3

minus2

minus1

minus4

minus3

minus2

]

]

(9)

Definition 18 Letting 119862 = 119862(1198671

119878 1198672

119878) be the pairwise

comparison matrix between 1198671

119878and 119867

2

119878 the preference

relations of 1198671119878and 119867

2

119878are defined as follows

119901 (1198671

119878gt 1198672

119878) =

10038161003816100381610038161003816sum119862119898119899gt0

119862119898119899

10038161003816100381610038161003816

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

119901 (1198671

119878= 1198672

119878) =

119862119898119899 = 0

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

119901 (1198671

119878lt 1198672

119878) =

10038161003816100381610038161003816sum119862119898119899lt0

119862119898119899

10038161003816100381610038161003816

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

(10)

It is obvious that 119901(1198671

119878gt 1198672

119878) + 119901(119867

1

119878= 1198672

119878) + 119901(119867

1

119878lt

1198672

119878) = 1 We say that 1198671

119878is superior to 119867

2

119878with the degree of

119901(1198671

119878gt 1198672

119878) denoted by 119867

1

119878≻119901(1198671

119878gt1198672

119878)1198672

119878 1198671119878is equal to 119867

2

119878

with the degree of 119901(1198671

119878= 1198672

119878) denoted by 119867

1

119878sim119901(1198671

119878=1198672

119878)1198672

119878

and 1198671

119878is inferior to 119867

2

119878with the degree of 119901(119867

1

119878lt 1198672

119878)

denoted by 1198671

119878≺119901(1198671

119878lt1198672

119878)1198672

119878

Considering Example 17 by Definition 18 (10) the pref-erence relations of 119867

1

119878and 119867

2

119878were calculated as 119901(119867

1

119878gt

1198672

119878) = 122 119901(119867

1

119878= 1198672

119878) = 222 and 119901(119867

1

119878lt 1198672

119878) = 1922

Thus the comparison results are1198671

119878≻122

1198672

1198781198671119878sim222

1198672

119878 and

1198671

119878≺1922

1198672

119878

33 Distance Measure of HFLTSs

Definition 19 Letting 119862 = 119862(1198671

119878 1198672

119878) be the pairwise

comparisonmatrix between1198671

119878and119867

2

119878 the distance between

1198671

119878and 119867

2

119878is defined as the average value of the pairwise

comparison matrix

119889 (1198671

119878 1198672

119878) =

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

|1198671

119878|

sum

119898=1

|1198672

119878|

sum

119899=1

119862119898119899(11)

Considering Example 17 one has 119889(1198671

119878 1198672

119878) = (minus18)

(3 times 4) = minus15

To preserve all the given information the discrete linguis-tic term set 119878 is extended to a continuous term set 119878 = 119904120572 |

120572 isin [minus119902 119902] where 119902 is a sufficiently large positive number If119904120572 isin 119878 then we call 119904120572 an original linguistic term otherwisewe call 119904120572 a virtual linguistic term

Remark 20 In general the decision-maker uses the originallinguistic terms to express hisher qualitative opinions andthe virtual linguistic terms can only appear in operations

Definition 21 The average value of anHFLTS119867119878 is defined as

Aver (119867119878) =

1

1003816100381610038161003816119867119878

1003816100381610038161003816

sum

119904119894isin119867119878

119904119894 =

1

1003816100381610038161003816119867119878

1003816100381610038161003816

sum

119904119894isin119867119878

Ind (119904119894) (12)

This definition is similar to the score function of an HFEDefinition 3

Considering Example 17 we have Aver(1198671119878) = 119904(1+2+3)3 =

1199042 = 2 and Aver(1198672119878) = 119904(2+3+4+5)4 = 11990435 =35

Theorem 22 Letting 119867119878 be an HFLTS on 119878 then0 le 119867119904minus le Aver (119867119878) le 119867119904+ le (|119878| minus 1) (13)

Proof It is straightforward and thus omitted

Theorem 23 Letting 1198671

119878and 119867

2

119878be two HFLTSs on 119878 the

distance between 1198671

119878and 119867

2

119878defined by the average value of

their pairwise comparison matrix is equal to the distance of thetwo average values of1198671

119878and119867

2

119878 that is the distance between

1198671

119878and 119867

2

119878can be easily obtained by

119889 (1198671

119878 1198672

119878) = Aver (1198671

119878) minus Aver (1198672

119878) (14)

Proof From Definitions 19 and 14 we have

119889 (1198671

119878 1198672

119878) =

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

|1198671

119878|

sum

119898=1

|1198672

119878|

sum

119899=1

119862119898119899

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119889 (119904119894 119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

(119904119894 minus 119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

( sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119904119894 minus sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

(

100381610038161003816100381610038161198672

119878

10038161003816100381610038161003816times sum

119904119894isin1198671119878

119904119894 minus

100381610038161003816100381610038161198671

119878

10038161003816100381610038161003816times sum

119904119895isin1198672119878

119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

119904119894 minus

1

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119895isin1198672119878

119904119895

= Aver (1198671119878) minus Aver (1198672

119878)

(15)

which completes the proof of Theorem 23

Mathematical Problems in Engineering 5

Considering Example 17 we have 119889(1198671

119878 1198672

119878) =

Aver(1198671119878) minus Aver(1198672

119878) = 2 minus 35 = minus15

By Theorem 23 we can easily obtain the following corol-lary

Corollary 24 Letting 1198671

119878 1198672119878and 119867

3

119878be three HFLTSs on S

then

(1) (|119878| minus 1) le 119889(1198671

119878 1198672

119878) le (|119878| minus 1)

(2) 119889(1198671

119878 1198672

119878) = minus119889(119867

2

119878 1198671

119878)

(3) 119889(1198671

119878 1198673

119878) = 119889(119867

1

119878 1198672

119878) + 119889(119867

2

119878 1198673

119878)

Proof They are straightforward and thus omitted

If 119889(1198671

119878 1198672

119878) gt 0 (or Aver(1198671

119878) gt Aver(1198672

119878)) then we

say that 1198671119878is superior to 119867

2

119878with the distance of 119889(1198671

119878 1198672

119878)

denoted by 1198671

119878

119889(1198671

1198781198672

119878)

≻ 1198672

119878 if 119889(1198671

119878 1198672

119878) = 0 (or Aver(1198671

119878) =

Aver(1198672119878)) then we say that 1198671

119878is indifferent to 119867

2

119878 denoted

by 1198671

119878sim 1198672

119878 if 119889(1198671

119878 1198672

119878) lt 0 (or Aver(1198671

119878) lt Aver(1198672

119878))

then we say that 1198671

119878is inferior to 119867

2

119878with the distance of

119889(1198671

119878 1198672

119878) denoted by 119867

1

119878

119889(1198671

1198781198672

119878)

≺ 1198672

119878

4 Multicriteria Decision-Making ModelsBased on Comparisons and DistanceMeasures of HFLTSs

In this section two new methods are presented for rankingand choice from a set of alternatives in the framework ofmulticriteria decision-making using linguistic informationOne is based on the comparisons and preference relationsof HFLTSs and the other is based on the distance measureof HFLTSs We adopt Example 5 in [56] (Example 25 in ourpaper) to illustrate the detailed processes of the twomethods

Example 25 ([see [56]) Let 119883 = 1199091 1199092 1199093 be a set ofalternatives 119862 = 1198881 1198882 1198883 a set of criteria defined for eachalternative and 119878 = 1199040 n (nothing) 1199041 vl (very low)1199042 l (low) 1199043 m (medium) 1199044 h (high) 1199045

vh (very high) 1199046 p (perfect) the linguistic term set that isused to generate the linguistic expressions The assessmentsthat are provided in such a problem are shown in Table 1 andthey are transformed into HFLTSs as shown in Table 2

41 Multicriteria Decision-Making Based onthe Comparisons of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe preference degrees between all the alternatives 119909119895 (119895 =

1 2 3)Considering criterion 1198881 119867

1199091

119878= 1199041 1199042 1199043 119867

1199092

119878=

1199042 1199043 and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees

about criterion 1198881 calculated using the comparisonmethod ofHFLTSs as described in Section 32 are 1199011198881(1199091 gt 1199092) = 171199011198881(1199091 = 1199092) = 27 1199011198881(1199091 lt 1199092) = 47 1199011198881(1199091 gt 1199093) =

027 1199011198881(1199091 = 1199093) = 027 1199011198881(1199091 lt 1199093) = 2727 1199011198881(1199092 gt

1199093) = 015 1199011198881(1199092 = 1199093) = 015 1199011198881(1199092 lt 1199093) = 1515

Table 1 Assessments that are provided for the decision problem

1198881

1198882

1198883

1199091

Between vl and m Between h and vh h1199092

Between l and m m Lower than l1199093

Greater than h Between vl and l Greater than h

Table 2 Assessments transformed into HFLTSs

1198881

1198882

1198883

1199091

1199041 1199042 1199043 119904

4 1199045 119904

4

1199092

1199042 1199043 119904

3 119904

0 1199041 1199042

1199093

1199044 1199045 1199046 119904

1 1199042 119904

4 1199045 1199046

Considering criterion 1198882 1198671199091

119878= 1199044 1199045 119867

1199092

119878= 1199043

and 1198671199093

119878= 1199041 1199042 so the preference degrees about criterion

1198882 calculated using the comparison method of HFLTSs asdescribed in Section 32 are 1199011198882(1199091 gt 1199092) = 33 1199011198882(1199091 =

1199092) = 03 1199011198882(1199091 lt 1199092) = 03 1199011198882(1199091 gt 1199093) = 12121199011198882(1199091 = 1199093) = 012 1199011198882(1199091 lt 1199093) = 012 1199011198882(1199092 gt 1199093) =

33 1199011198882(1199092 = 1199093) = 03 1199011198882(1199092 lt 1199093) = 03Considering criterion 1198883 119867

1199091

119878= 1199044 119867

1199092

119878= 1199040 1199041 1199042

and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees about

criterion 1198883 calculated using the comparison method ofHFLTSs as described in Section 32 are 1199011198883(1199091 gt 1199092) = 991199011198883(1199091 = 1199092) = 09 1199011198883(1199091 lt 1199092) = 09 1199011198883(1199091 gt 1199093) = 041199011198883(1199091 = 1199093) = 14 1199011198883(1199091 lt 1199093) = 34 1199011198883(1199092 gt 1199093) =

036 1199011198883(1199092 = 1199093) = 036 1199011198883(1199092 lt 1199093) = 3636

Step 2 Aggregate the preference relations using the weightedaverage method 119901(119909119895 gt 119909119896) = sum(119908119894 times 119901119888119894(119909119895 gt 119909119896))119901(119909119895 = 119909119896) = sum(119908119894 times 119901119888i(119909119895 = 119909119896)) and 119901(119909119895 lt 119909119896) =

sum(119908119894 times 119901119888119894(119909119895 lt 119909119896)) where 119908119894 is the weight of criterion 119888119894and sum(119908119894) = 1 In this paper 119908119894 = 13 119894 = 1 2 3 Thusthe final preference relations are (1199091 gt 1199092) = 1521 119901(1199091 =

1199092) = 221 119901(1199091 lt 1199092) = 421 119901(1199091 gt 1199093) = 13119901(1199091 = 1199093) = 112 119901(1199091 lt 1199093) = 712 119901(1199092 gt 1199093) = 13119901(1199092 = 1199093) = 0 119901(1199092 lt 1199093) = 23

Step 3 Rank the alternatives using the nondominance choicedegree method as described in [56] From the results ofStep 2 it can be easily obtained that

119875119878

119863= (

minus

11

21

0

0 minus 0

1

4

1

3

minus

) (16)

Thus NDD1 = min(1 minus 0) (1 minus 14) = 34 NDD2 =

min(1 minus 1121) (1 minus 13) = 1021 and NDD3 = min(1 minus

0) (1 minus 0) = 1 Finally the ranking of alternatives is 1199093 gt

1199091 gt 1199092

6 Mathematical Problems in Engineering

Table 3 Average values of the assessments

1198881

1198882

1198883

1199091

2 45 41199092

25 3 11199093

5 15 5

Table 4 Aggregation results of each alternative

1199091

1199092

1199093

Aggregation result 35 22 38

42 Multicriteria Decision-Making Based onthe Distance Measures of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe average values of HFLTSs for all the alternatives 119909119895 (119895 =

1 2 3) The results are shown in Table 3

Step 2 Aggregate the average values using the weightedaverage method The results are shown in Table 4

Step 3 Rank the alternatives using the distance measuremethod Thus the ranking of alternatives is 1199093

03

≻ 1199091

13

≻ 1199092

43 Results Analysis In [56] the ranking of alternatives is1199091 ≻ 1199093 ≻ 1199092 while both methods in this paper are1199093 ≻ 1199091 ≻ 1199092 Note that the practical decision-makingproblem is quite different from other applications where well-establishedmeasures can be used to quantify the performancefor validation In decision-making usually there is no groundtruth data or quantitativemeasures to assess the performanceof amethod [37]This is why ldquoplausibilityrdquo is used rather thanldquovalidationrdquo Here we analyze the original assessments abouteach criterion of alternatives 1199091 and 1199093 Considering criterion1198881 the original assessments of 1199091 and 1199093 are ldquobetween vland mrdquo and ldquogreater than hrdquo respectively so it is obviously1199093 ≻ 1199091 about criterion 1198881 Considering criterion 1198882 theoriginal assessments of 1199091 and 1199093 are ldquobetween h and vhrdquoand ldquobetween vl and lrdquo respectively so this time 1199091 ≻ 1199093Considering criterion 1198883 the original assessments of 1199091 and1199093 are ldquohrdquo and ldquogreater than hrdquo respectively so 1199093 ≻ 1199091 againSummarily 1199093 ≻ 1199091 occurs twice while 1199091 ≻ 1199093 only onceThus we believe that our result is more plausible

5 Conclusion

The comparison and distance measure of HFLTSs are fun-damentally important in many decision-making problemsunder hesitant fuzzy linguistic environments From an exam-ple we found that there existed two defects when comparingHFLTSs using the previous preference degree method Byanalyzing the definition of an HFLTS a new comparisonmethod based on pairwise comparisons of each linguisticterm in the two HFLTSs has been put forward This compar-ison method does not need the assumption that the valuesin all HFLTSs are arranged in an increasing order and two

HFLTSs have the same length when comparing them Thenwe have defined a distancemeasuremethod betweenHFLTSsbased on pairwise comparisons Further we have proved thatthis distance is equal to the distance of the average values ofHFLTSs which makes the distance measure much simplerFinally two new methods for multicriteria decision-makingin which experts provide their assessments by HFLTSs havebeen proposedThe encouraging results demonstrate that ourmethods in this paper are more reasonable

In the future the application of HFLTSs to groupdecision-making problems will be explored We will alsoinvestigate how to obtain the weights of criteria underhesitant fuzzy linguistic environments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[2] M Mizumoto and K Tanaka ldquoSome properties of fuzzy sets oftype 2rdquo Information andControl vol 31 no 4 pp 312ndash340 1976

[3] J M Mendel and R I B John ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[4] J M Mendel and H Wu ldquoType-2 fuzzistics for symmetricinterval type-2 fuzzy sets part 1 forward problemsrdquo IEEETransactions on Fuzzy Systems vol 14 no 6 pp 781ndash792 2006

[5] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[6] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[7] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[8] K T Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[9] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[10] H Bustince and P Burillo ldquoVague sets are intuitionistic fuzzysetsrdquo Fuzzy Sets and Systems vol 79 no 3 pp 403ndash405 1996

[11] R R Yager ldquoOn the theory of bagsrdquo International Journal ofGeneral Systems vol 13 no 1 pp 23ndash37 1986

[12] S Miyamoto ldquoRemarks on basics of fuzzy sets and fuzzymultisetsrdquo Fuzzy Sets and Systems vol 156 no 3 pp 427ndash4312005

[13] J M Garibaldi M Jaroszewski and S Musikasuwan ldquoNonsta-tionary fuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 16no 4 pp 1072ndash1086 2008

[14] D Li J Han X Shi and M C Chan ldquoKnowledge representa-tion and discovery based on linguistic atomsrdquoKnowledge-BasedSystems vol 10 no 7 pp 431ndash440 1998

[15] D Li C Liu andWGan ldquoAnew cognitivemodel cloudmodelrdquoInternational Journal of Intelligent Systems vol 24 no 3 pp357ndash375 2009

Mathematical Problems in Engineering 7

[16] X Yang L Zeng F Luo and S Wang ldquoCloud hierarchicalanalysisrdquo Journal of Information amp Computational Science vol7 no 12 pp 2468ndash2477 2010

[17] X Yang L Zeng and R Zhang ldquoCloud Delphi methodrdquoInternational Journal of Uncertainty Fuzziness and Knowlege-Based Systems vol 20 no 1 pp 77ndash97 2012

[18] X Yang L Yan and L Zeng ldquoHow to handle uncertaintiesin AHP the Cloud Delphi hierarchical analysisrdquo InformationSciences vol 222 pp 384ndash404 2013

[19] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[20] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju Island Korea August 2009

[21] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 3 pp 199ndash249 1975

[22] L A Zadeh ldquoThe concept of a linguistic variable and its appli-cation to approximate reasoningmdashIIrdquo Information Sciences vol4 pp 301ndash357 1975

[23] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIIIrdquo Information Sciences vol9 no 1 pp 43ndash80 1975

[24] J Ma D Ruan Y Xu and G Zhang ldquoA fuzzy-set approach totreat determinacy and consistency of linguistic terms in multi-criteria decision makingrdquo International Journal of ApproximateReasoning vol 44 no 2 pp 165ndash181 2007

[25] G Bordogna and G Pasi ldquoA fuzzy linguistic approach general-izing boolean information retrieval amodel and its evaluationrdquoJournal of the American Society for Information Science vol 44no 2 pp 70ndash82 1993

[26] G Bordogna and G Pasi ldquoAn ordinal information retrievalmodelrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 9 supplement 1 2001

[27] J Kacprzyk and S Zadrozny ldquoLinguistic database summariesand their protoforms towards natural language based knowl-edge discovery toolsrdquo Information Sciences vol 173 no 4 pp281ndash304 2005

[28] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[29] H Ishibuchi T Nakashima and M Nii Classification andModeling with Linguistic Information Granules AdvancedApproaches to Linguistic Data Mining Springer Berlin Ger-many 2004

[30] R Degani and G Bortolan ldquoThe problem of linguistic approx-imation in clinical decision makingrdquo International Journal ofApproximate Reasoning vol 2 no 2 pp 143ndash162 1988

[31] E Sanchez ldquoTruth-qualification and fuzzy relations in naturallanguages application to medical diagnosisrdquo Fuzzy Sets andSystems vol 84 no 2 pp 155ndash167 1996

[32] H Lee ldquoApplying fuzzy set theory to evaluate the rate ofaggregative risk in software developmentrdquo Fuzzy Sets andSystems vol 79 no 3 pp 323ndash336 1996

[33] L Martınez ldquoSensory evaluation based on linguistic decisionanalysisrdquo International Journal of Approximate Reasoning vol44 no 2 pp 148ndash164 2007

[34] L Martınez J Liu D Ruan and J Yang ldquoDealing with het-erogeneous information in engineering evaluation processesrdquoInformation Sciences vol 177 no 7 pp 1533ndash1542 2007

[35] J Lu Y Zhu X Zeng L Koehl J Ma and G Zhang ldquoA linguis-tic multi-criteria group decision support system for fabric handevaluationrdquo FuzzyOptimization andDecisionMaking vol 8 no4 pp 395ndash413 2009

[36] R de Andres M Espinilla and L Martınez ldquoAn extendedhierarchical linguistic model for managing integral evaluationrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 486ndash500 2010

[37] D Wu and J M Mendel ldquoComputing with words for hierar-chical decision making applied to evaluating a weapon systemrdquoIEEE Transactions on Fuzzy Systems vol 18 no 3 pp 441ndash4602010

[38] F Herrera and E Herrera-Viedma ldquoChoice functions andmechanisms for linguistic preference relationsrdquo European Jour-nal of Operational Research vol 120 no 1 pp 144ndash161 2000

[39] F Herrera E Herrera-Viedma and L Martınez ldquoA fusionapproach for managingmulti-granularity linguistic term sets indecision makingrdquo Fuzzy Sets and Systems vol 114 no 1 pp 43ndash58 2000

[40] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[41] B Arfi ldquoFuzzy decisionmaking in politics a linguistic fuzzy-setapproachrdquo Political Analysis vol 13 no 1 pp 23ndash56 2005

[42] V Huynh and Y Nakamori ldquoA satisfactory-oriented approachto multiexpert decision-making with linguistic assessmentsrdquoIEEE Transactions on Systems Man and Cybernetics B Cyber-netics vol 35 no 2 pp 184ndash196 2005

[43] D Ben-Arieh and Z Chen ldquoLinguistic group decision-makingopinion aggregation and measures of consensusrdquo Fuzzy Opti-mization and Decision Making vol 5 no 4 pp 371ndash386 2006

[44] J L Garcıa-Lapresta ldquoA general class of simple majority deci-sion rules based on linguistic opinionsrdquo Information Sciencesvol 176 no 4 pp 352ndash365 2006

[45] J Wang and J Hao ldquoFuzzy linguistic PERTrdquo IEEE Transactionson Fuzzy Systems vol 15 no 2 pp 133ndash144 2007

[46] T C Wang and Y H Chen ldquoApplying fuzzy linguistic prefer-ence relations to the improvement of consistency of fuzzy AHPrdquoInformation Sciences vol 178 no 19 pp 3755ndash3765 2008

[47] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[48] P D Liu ldquoA novel method for hybridmultiple attribute decisionmakingrdquo Knowledge-Based Systems vol 22 no 5 pp 388ndash3912009

[49] Y Dong W Hong Y Xu and S Yu ldquoSelecting the individualnumerical scale and prioritization method in the analytichierarchy process a 2-Tuple fuzzy linguistic approachrdquo IEEETransactions on Fuzzy Systems vol 19 no 1 pp 13ndash25 2011

[50] L A Zadeh ldquoFuzzy logic = computing with wordsrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 103ndash111 1996

[51] J M Mendel ldquoFuzzy sets for words a new beginningrdquo inProceedings of the IEEE International conference on FuzzySystems pp 37ndash42 St Louis Mo USA May 2003

[52] J MMendel ldquoComputing with words and its relationships withfuzzisticsrdquo Information Sciences vol 177 no 4 pp 988ndash10062007

[53] F Liu and J M Mendel ldquoEncoding words into interval type-2 fuzzy sets using an interval approachrdquo IEEE Transactions onFuzzy Systems vol 16 no 6 pp 1503ndash1521 2008

8 Mathematical Problems in Engineering

[54] S Coupland J M Mendel and D Wu ldquoEnhanced intervalapproach for encoding words into interval type-2 fuzzy sets andconvergence of the word FOUsrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10) pp1ndash8 Barcelona Spain July 2010

[55] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[56] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[57] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[58] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[59] W Wang and X Xin ldquoDistance measure between intuitionisticfuzzy setsrdquo Pattern Recognition Letters vol 26 no 13 pp 2063ndash2069 2005

[60] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[61] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[62] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[63] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[64] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[65] Y Xu and H Wang ldquoDistance measure for linguistic decisionmakingrdquo Systems Engineering Procedia vol 1 pp 450ndash456 2011

[66] Y M Wang J B Yang and D L Xu ldquoA preference aggregationmethod through the estimation of utility intervalsrdquo Computersand Operations Research vol 32 no 8 pp 2027ndash2049 2005

[67] H Liao Z Xu and X Zeng ldquoDistance and similarity measuresfor hesitant fuzzy linguistic term sets and their application inmulti-criteria decision makingrdquo Information Sciences vol 271pp 125ndash142 2014

[68] B Zhu Z Xu andMXia ldquoHesitant fuzzy geometric Bonferronimeansrdquo Information Sciences vol 205 pp 72ndash85 2012

[69] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[70] X Gu Y Wang and B Yang ldquoA method for hesitant fuzzymultiple attribute decision making and its application to riskinvestmentrdquo Journal of Convergence Information Technologyvol 6 no 6 pp 282ndash287 2011

[71] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013

[72] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012

[73] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

an expert hesitating among several values for a linguistic vari-able For example when people assess a qualitative criterionthey prefer to use a linguistic one such as ldquobetween mediumand very highrdquo which contains several linguistic terms119898119890119889119894119906119898 ℎ119894119892ℎ V119890119903119910ℎ119894119892ℎ rather than a single linguisticterm In practical decision-making process uncertainty andhesitancy are usually unavoidable problemsTheHFLTSs candeal with such uncertainty and hesitancy more objectivelyand thus it is very necessary to develop some theories aboutHFLTSs

Comparisons and distance measures used for measuringthe deviations of different arguments are fundamentallyimportant in a variety of applications In the existing litera-ture there are a number of studies on distance measures forintuitionistic fuzzy sets [57ndash60] interval-valued intuitionisticfuzzy sets [61] hesitant fuzzy sets [62 63] linguistic values[64 65] and so forth Nevertheless an HFLTS is a linguisticterm subset and the comparison among these elements isnot simple In [56] Rodriguez et al introduced the conceptof envelope for an HFLTS and then ranked HFLTSs usingthe preference degree method of interval values [66] Butbecause an HFLTS is a set of discrete linguistic terms itmay seem problematical using the preference degree methodfor continuous interval to compare these discrete terms ofHFLTSs Up to now just a few research has been done onthe distance measure of HFLTSs [67] Consequently it isvery necessary to develop some comparison methods anddistance measure methods for HFLTSs In [67] to calculatethe distance of two HFLTSs Liao et al extend the shorterHFLTS by adding any value in it until it has the same length ofthe longer one according to the decision-makerrsquos preferencesand actual situations In this paper we present a new com-parison method of HFLTSs based on pairwise comparisonsof each linguistic term in the two HFLTSs Then a distancemeasure method based on the pairwise comparison matrixof HFLTSs is proposed without adding any value Finallywe utilize the comparison method and distance measuremethod to develop some approaches to solve themulticriteriadecision-making problems under hesitant fuzzy linguisticenvironments

The rest of the paper is organized as follows In Section 2the concepts of hesitant fuzzy sets and HFLTSs are intro-duced also the defects of the previous comparison methodfor HFLTSs are analyzed according to an example Section 3describes the comparison and distance measure of HFLTSsbased on the proposed pairwise comparison method InSection 4 a multicriteria decision-making problem is shownto illustrate the detailed processes and effectiveness of tworanking methods which are based on the comparisons anddistance measures of HFLTSs respectively Finally Section 5draws our conclusions and presents suggestions for futureresearch

2 Preliminaries

21 Hesitant Fuzzy Sets Hesitant fuzzy sets (HFSs) were firstintroduced by Torra [19] and Torra and Narukawa [20] Themotivation is that when determining the membership degree

of an element into a set the difficulty is not because we havea margin of error (such as an interval) but because we haveseveral possible values

Definition 1 (see [19]) Let 119883 be a fixed set a hesitant fuzzyset (HFS) on 119883 is in terms of a function ℎ that when appliedto 119883 returns a subset of [0 1]

To be easily understood Zhu et al [68] represented theHFS as the following mathematical symbol

119864 = ⟨119909 ℎ (119909)⟩ | 119909 isin 119883 (1)

where ℎ(119909) is a set of some values in [0 1] denoting thepossible membership degrees of the element 119909 isin 119883 to the set119864 Liao et al [67] called ℎ(119909) a hesitant fuzzy element (HFE)

Example 2 Let ℎ = 02 03 04 then ℎ is an HFE

Definition 3 (see [69]) For an HFE ℎ the score function of ℎis defined as

119904 (ℎ) =

1

ℎsum

120574isinℎ

120574 (2)

where ℎ is the number of the elements in ℎ

For two HFEs ℎ1 and ℎ2 if 119904(ℎ1) gt 119904(ℎ2) then ℎ1 issuperior to ℎ2 denoted by ℎ1 ≻ ℎ2 if 119904(ℎ1) = 119904(ℎ2) thenℎ1 is indifferent with ℎ2 denoted by ℎ1 sim ℎ2

Example 4 Assume that we have three HFEs ℎ1 =

02 03 04 ℎ2 = 02 035 05 and ℎ3 = 03 04 thenaccording to the score function of HFE (2) and Definition 3we have 119904(ℎ1) = (02 + 03 + 04)3 = 03 119904(ℎ2) = (02 +

035 + 05)3 = 035 and 119904(ℎ3) = (03 + 04)2 = 035 Thus119904(ℎ2) = 119904(ℎ3) gt 119904(ℎ1) that is the ranking is ℎ2 sim ℎ3 ≻ ℎ1

The concept of HFS is very useful to express peoplersquoshesitancy in daily life So since it was introduced more andmore decision-making theories and methods under hesitantfuzzy environment have been developed [56 62 63 67ndash73]

22 Hesitant Fuzzy Linguistic Term Sets Similarly to theHFSan expertmay hesitate among several linguistic terms such asldquobetween medium and very highrdquo or ldquolower than mediumrdquoto assess a qualitative linguistic variable To deal with suchsituations Rodriguez et al [56] introduced the concept ofhesitant fuzzy linguistic term sets (HFLTSs)

Definition 5 (see [56]) Suppose that 119878 = 1199040 119904119892 is afinite and totally ordered discrete linguistic term set where119904119894 represents a possible value for a linguistic variable AnHFLTS 119867119878 is defined as an ordered finite subset of theconsecutive linguistic terms of 119878

It is required that the linguistic term set 119878 should satisfythe following characteristics

(1) the set is ordered 119904119894 gt 119904119895 if and only if 119894 gt 119895(2) there is a negation operator Neg(119904119894) = 119904119892minus119894

Mathematical Problems in Engineering 3

Example 6 Let 119878 be a linguistic term set 119878 = 1199040

n (nothing) 1199041 vl (very low) 1199042 l (low) 1199043 m (medium)1199044 h (high) 1199045 vh (very high) and 1199046 p (perfect) twodifferent HFLTSs might be 119867

1

119878= 1199041 vl 1199042 l 1199043 m and

1198672

119878= 1199043 m 1199044 h

Definition 7 One defines the number of linguistic terms inthe HFLTS 119867119878 as the cardinality of 119867119878 denoted by |119867119878| InExample 6 |1198671

119878| = 3 and |119867

2

119878| = 2

Definition 8 (see [56]) The lower bound 119867119878minus and upperbound 119867119878+ of the HFLTS 119867119878 are defined as 119867119878minus = min119904119894 |119904119894 isin 119867119878 and 119867119878+ = max119904119894 | 119904119894 isin 119867119878

Definition 9 (see [56]) The envelope of the HFLTS 119867119878env(119867119878) is defined as the linguistic interval [119867119878minus 119867119904+] =

[Ind(119867119878minus) Ind(119867119878+)] where Ind provides the index of thelinguistic term that is Ind(119904119894) = 119894 In Example 6 env(1198671

119878) =

[1199041 1199043] = [1 3] and env(1198672119878) = [1199043 1199044] = [3 4]

Based on the definition of envelope Rodriguez et al [56]compare twoHFLTSs using the comparisonmethod betweentwo numerical intervals introduced by Wang et al [66]

Definition 10 (see [66]) Letting 119860 = [1198861 1198862] and 119861 = [1198871 1198872]

be two intervals the preference degree of119860 over 119861 (or119860 gt 119861)is defined as

119875 (119860 gt 119861) =

max (0 1198862 minus 1198871) minus max (0 1198861 minus 1198872)

(1198862 minus 1198861) + (1198872 minus 1198871)

(3)

and the preference degree of 119861 over 119860 (or 119861 gt 119860) is definedas

119875 (119861 gt 119860) =

max (0 1198872 minus 1198861) minus max (0 1198871 minus 1198862)

(1198862 minus 1198861) + (1198872 minus 1198871)

(4)

Example 11 Let 1198671

119878= 1199041 1199042 1199043 119867

2

119878= 1199043 1199044 and

1198673

119878= 1199045 1199046 be three different HFLTSs on 119878 According to

Definition 9 we have env(1198672119878) = [1 3] env(1198672

119878) = [3 4]

and env(1198673119878) = [5 6] The preference degrees calculated by

Definition 10 (3) and (4) are

119875 (1198671

119878gt 1198672

119878) = 0 119875 (119867

2

119878gt 1198671

119878) = 1

119875 (1198671

119878gt 1198673

119878) = 0 119875 (119867

3

119878gt 1198671

119878) = 1

119875 (1198672

119878gt 1198673

119878) = 0 119875 (119867

3

119878gt 1198672

119878) = 1

(5)

From Example 11 mentioned above it can be observedthat when we compare two HFLTSs using the preferencedegree method there exist two defects as follows

(1) The result 119875(1198672

119878gt 119867

1

119878) = 1 indicates that 119867

2

119878

is absolutely superior to 1198671

119878 In fact both 119867

1

119878and 119867

2

119878

contain the linguistic term 1199043 It means that the value of alinguistic variable may be equal in these two cases Thus itis unreasonable to say that 1198672

119878is absolutely superior to 119867

1

119878

(2) The result 119875(1198673

119878gt 1198671

119878) = 119875(119867

2

119878gt 1198671

119878) = 1 meaning

that when comparedwith1198671

119878 the twoHFLTSs1198672

119878and119867

3

119878are

identical In fact 1198673119878is more superior to 119867

1

119878compared to 119867

2

119878

to 1198671

119878 Thus using the preference degree method to compare

HFLTSs may result in losing some important informationBased on the analysis mentioned above we think that it

is not suitable to compare discrete linguistic terms in HFLTSsusing the comparison method for continuous numericalintervals By the definition of an HFLTS we know thatevery linguistic term in it is a possible value of the linguisticinformation And noting that the twoHFLTSs for comparingmay have different lengths So when comparing twoHFLTSsit needs pairwise comparisons of each linguistic term in them

3 Comparison and DistanceMeasure of HFLTSs

31 Distance between Two Single Linguistic Terms Let 119904119894 119904119895 isin119878 be two linguistic terms Xu [64] defined the deviationmeasure between 119904119894 and 119904119895 as follows

119889 (119904119894 119904119895) =

1003816100381610038161003816119894 minus 119895

1003816100381610038161003816

119879

(6)

where 119879 is the cardinality of 119878 that is 119879 = |119878|If only one preestablished linguistic term set 119878 is used in

a decision-making model we can simply consider [49 65]119889(119904119894 119904119895) = |119894 minus 119895| = 119904|119894minus119895|

Definition 12 Letting 119904119894 119904119895 isin 119878 be two single linguistic termsthen we call

119889 (119904119894 119904119895) = 119904119894 minus 119904119895 = 119894 minus 119895 (7)

the distance between 119904119894 and 119904119895

The distance measure between 119904119894 and 119904119895 has a definitephysical implication and reflects the relative position anddistance between 119904119894 and 119904119895 If 119889(119904119894 119904119895) = 0 then 119904119894 = 119904119895 If119889(119904119894 119904119895) gt 0 then 119904119894 gt 119904119895 If 119889(119904119894 119904119895) lt 0 then 119904119894 lt 119904119895

Theorem13 Letting 119904119894 119904119895 119904119896 isin 119878 be three linguistic terms then(1) 119889(119904119894 119904119895) = minus119889(119904119895 119904119894)(2) (|119878| minus 1) le 119889(119904119894 119904119895) le (|119878| minus 1)(3) 119889(119904119894 119904119896) = 119889(119904119894 119904119895) + 119889(119904119895 119904119896)

Proof They are straightforward and thus omitted

32 Comparison of HFLTSs The comparison of HFLTSs isnecessary in many problems such as ranking and selectionHowever anHFLTS is a linguistic term subset which containsseveral linguistic terms and the comparison among HFLTSsis not simple Here a new comparison method of HFLTSswhich is based on pairwise comparisons of each linguisticterm in the two HFLTSs is put forward

Definition 14 Letting 1198671

119878and 119867

2

119878be two HFLTSs on 119878 then

one defines the pairwise comparison matrix between1198671

119878and

1198672

119878as follows

119862 (1198671

119878 1198672

119878) = [119889 (119904119894 119904119895)]|1198671

119878|times|1198672119878| 119904119894 isin 119867

1

119878 119904119895 isin 119867

2

119878 (8)

4 Mathematical Problems in Engineering

Remark 15 The number of linguistic terms in the twoHFLTSs 1198671

119878and 119867

2

119878 may be unequal that is |1198671

119878| = |119867

2

119878|

To deal with such situations usually it is necessary to extendthe shorter one by adding the stated value several times init [62 63] while our pairwise comparison method does notrequire this step

Remark 16 From Definition 14 we have [119862(1198671

119878 1198672

119878)] =

minus[119862(1198672

119878 1198671

119878)]

119879 where 119879 is the transpose operator of matrix

Example 17 Let 1198671119878

= 1199041 1199042 1199043 and 1198672

119878= 1199042 1199043 1199044 1199045 be

twoHFLTSs on 119878 According toDefinition 14 the comparisonmatrix 119862 between 119867

1

119878and 119867

2

119878is

1199041

1199042

1199043

1199042 1199043 1199044 1199045

[

[

minus1

0

1

minus2

minus1

0

minus3

minus2

minus1

minus4

minus3

minus2

]

]

(9)

Definition 18 Letting 119862 = 119862(1198671

119878 1198672

119878) be the pairwise

comparison matrix between 1198671

119878and 119867

2

119878 the preference

relations of 1198671119878and 119867

2

119878are defined as follows

119901 (1198671

119878gt 1198672

119878) =

10038161003816100381610038161003816sum119862119898119899gt0

119862119898119899

10038161003816100381610038161003816

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

119901 (1198671

119878= 1198672

119878) =

119862119898119899 = 0

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

119901 (1198671

119878lt 1198672

119878) =

10038161003816100381610038161003816sum119862119898119899lt0

119862119898119899

10038161003816100381610038161003816

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

(10)

It is obvious that 119901(1198671

119878gt 1198672

119878) + 119901(119867

1

119878= 1198672

119878) + 119901(119867

1

119878lt

1198672

119878) = 1 We say that 1198671

119878is superior to 119867

2

119878with the degree of

119901(1198671

119878gt 1198672

119878) denoted by 119867

1

119878≻119901(1198671

119878gt1198672

119878)1198672

119878 1198671119878is equal to 119867

2

119878

with the degree of 119901(1198671

119878= 1198672

119878) denoted by 119867

1

119878sim119901(1198671

119878=1198672

119878)1198672

119878

and 1198671

119878is inferior to 119867

2

119878with the degree of 119901(119867

1

119878lt 1198672

119878)

denoted by 1198671

119878≺119901(1198671

119878lt1198672

119878)1198672

119878

Considering Example 17 by Definition 18 (10) the pref-erence relations of 119867

1

119878and 119867

2

119878were calculated as 119901(119867

1

119878gt

1198672

119878) = 122 119901(119867

1

119878= 1198672

119878) = 222 and 119901(119867

1

119878lt 1198672

119878) = 1922

Thus the comparison results are1198671

119878≻122

1198672

1198781198671119878sim222

1198672

119878 and

1198671

119878≺1922

1198672

119878

33 Distance Measure of HFLTSs

Definition 19 Letting 119862 = 119862(1198671

119878 1198672

119878) be the pairwise

comparisonmatrix between1198671

119878and119867

2

119878 the distance between

1198671

119878and 119867

2

119878is defined as the average value of the pairwise

comparison matrix

119889 (1198671

119878 1198672

119878) =

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

|1198671

119878|

sum

119898=1

|1198672

119878|

sum

119899=1

119862119898119899(11)

Considering Example 17 one has 119889(1198671

119878 1198672

119878) = (minus18)

(3 times 4) = minus15

To preserve all the given information the discrete linguis-tic term set 119878 is extended to a continuous term set 119878 = 119904120572 |

120572 isin [minus119902 119902] where 119902 is a sufficiently large positive number If119904120572 isin 119878 then we call 119904120572 an original linguistic term otherwisewe call 119904120572 a virtual linguistic term

Remark 20 In general the decision-maker uses the originallinguistic terms to express hisher qualitative opinions andthe virtual linguistic terms can only appear in operations

Definition 21 The average value of anHFLTS119867119878 is defined as

Aver (119867119878) =

1

1003816100381610038161003816119867119878

1003816100381610038161003816

sum

119904119894isin119867119878

119904119894 =

1

1003816100381610038161003816119867119878

1003816100381610038161003816

sum

119904119894isin119867119878

Ind (119904119894) (12)

This definition is similar to the score function of an HFEDefinition 3

Considering Example 17 we have Aver(1198671119878) = 119904(1+2+3)3 =

1199042 = 2 and Aver(1198672119878) = 119904(2+3+4+5)4 = 11990435 =35

Theorem 22 Letting 119867119878 be an HFLTS on 119878 then0 le 119867119904minus le Aver (119867119878) le 119867119904+ le (|119878| minus 1) (13)

Proof It is straightforward and thus omitted

Theorem 23 Letting 1198671

119878and 119867

2

119878be two HFLTSs on 119878 the

distance between 1198671

119878and 119867

2

119878defined by the average value of

their pairwise comparison matrix is equal to the distance of thetwo average values of1198671

119878and119867

2

119878 that is the distance between

1198671

119878and 119867

2

119878can be easily obtained by

119889 (1198671

119878 1198672

119878) = Aver (1198671

119878) minus Aver (1198672

119878) (14)

Proof From Definitions 19 and 14 we have

119889 (1198671

119878 1198672

119878) =

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

|1198671

119878|

sum

119898=1

|1198672

119878|

sum

119899=1

119862119898119899

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119889 (119904119894 119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

(119904119894 minus 119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

( sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119904119894 minus sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

(

100381610038161003816100381610038161198672

119878

10038161003816100381610038161003816times sum

119904119894isin1198671119878

119904119894 minus

100381610038161003816100381610038161198671

119878

10038161003816100381610038161003816times sum

119904119895isin1198672119878

119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

119904119894 minus

1

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119895isin1198672119878

119904119895

= Aver (1198671119878) minus Aver (1198672

119878)

(15)

which completes the proof of Theorem 23

Mathematical Problems in Engineering 5

Considering Example 17 we have 119889(1198671

119878 1198672

119878) =

Aver(1198671119878) minus Aver(1198672

119878) = 2 minus 35 = minus15

By Theorem 23 we can easily obtain the following corol-lary

Corollary 24 Letting 1198671

119878 1198672119878and 119867

3

119878be three HFLTSs on S

then

(1) (|119878| minus 1) le 119889(1198671

119878 1198672

119878) le (|119878| minus 1)

(2) 119889(1198671

119878 1198672

119878) = minus119889(119867

2

119878 1198671

119878)

(3) 119889(1198671

119878 1198673

119878) = 119889(119867

1

119878 1198672

119878) + 119889(119867

2

119878 1198673

119878)

Proof They are straightforward and thus omitted

If 119889(1198671

119878 1198672

119878) gt 0 (or Aver(1198671

119878) gt Aver(1198672

119878)) then we

say that 1198671119878is superior to 119867

2

119878with the distance of 119889(1198671

119878 1198672

119878)

denoted by 1198671

119878

119889(1198671

1198781198672

119878)

≻ 1198672

119878 if 119889(1198671

119878 1198672

119878) = 0 (or Aver(1198671

119878) =

Aver(1198672119878)) then we say that 1198671

119878is indifferent to 119867

2

119878 denoted

by 1198671

119878sim 1198672

119878 if 119889(1198671

119878 1198672

119878) lt 0 (or Aver(1198671

119878) lt Aver(1198672

119878))

then we say that 1198671

119878is inferior to 119867

2

119878with the distance of

119889(1198671

119878 1198672

119878) denoted by 119867

1

119878

119889(1198671

1198781198672

119878)

≺ 1198672

119878

4 Multicriteria Decision-Making ModelsBased on Comparisons and DistanceMeasures of HFLTSs

In this section two new methods are presented for rankingand choice from a set of alternatives in the framework ofmulticriteria decision-making using linguistic informationOne is based on the comparisons and preference relationsof HFLTSs and the other is based on the distance measureof HFLTSs We adopt Example 5 in [56] (Example 25 in ourpaper) to illustrate the detailed processes of the twomethods

Example 25 ([see [56]) Let 119883 = 1199091 1199092 1199093 be a set ofalternatives 119862 = 1198881 1198882 1198883 a set of criteria defined for eachalternative and 119878 = 1199040 n (nothing) 1199041 vl (very low)1199042 l (low) 1199043 m (medium) 1199044 h (high) 1199045

vh (very high) 1199046 p (perfect) the linguistic term set that isused to generate the linguistic expressions The assessmentsthat are provided in such a problem are shown in Table 1 andthey are transformed into HFLTSs as shown in Table 2

41 Multicriteria Decision-Making Based onthe Comparisons of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe preference degrees between all the alternatives 119909119895 (119895 =

1 2 3)Considering criterion 1198881 119867

1199091

119878= 1199041 1199042 1199043 119867

1199092

119878=

1199042 1199043 and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees

about criterion 1198881 calculated using the comparisonmethod ofHFLTSs as described in Section 32 are 1199011198881(1199091 gt 1199092) = 171199011198881(1199091 = 1199092) = 27 1199011198881(1199091 lt 1199092) = 47 1199011198881(1199091 gt 1199093) =

027 1199011198881(1199091 = 1199093) = 027 1199011198881(1199091 lt 1199093) = 2727 1199011198881(1199092 gt

1199093) = 015 1199011198881(1199092 = 1199093) = 015 1199011198881(1199092 lt 1199093) = 1515

Table 1 Assessments that are provided for the decision problem

1198881

1198882

1198883

1199091

Between vl and m Between h and vh h1199092

Between l and m m Lower than l1199093

Greater than h Between vl and l Greater than h

Table 2 Assessments transformed into HFLTSs

1198881

1198882

1198883

1199091

1199041 1199042 1199043 119904

4 1199045 119904

4

1199092

1199042 1199043 119904

3 119904

0 1199041 1199042

1199093

1199044 1199045 1199046 119904

1 1199042 119904

4 1199045 1199046

Considering criterion 1198882 1198671199091

119878= 1199044 1199045 119867

1199092

119878= 1199043

and 1198671199093

119878= 1199041 1199042 so the preference degrees about criterion

1198882 calculated using the comparison method of HFLTSs asdescribed in Section 32 are 1199011198882(1199091 gt 1199092) = 33 1199011198882(1199091 =

1199092) = 03 1199011198882(1199091 lt 1199092) = 03 1199011198882(1199091 gt 1199093) = 12121199011198882(1199091 = 1199093) = 012 1199011198882(1199091 lt 1199093) = 012 1199011198882(1199092 gt 1199093) =

33 1199011198882(1199092 = 1199093) = 03 1199011198882(1199092 lt 1199093) = 03Considering criterion 1198883 119867

1199091

119878= 1199044 119867

1199092

119878= 1199040 1199041 1199042

and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees about

criterion 1198883 calculated using the comparison method ofHFLTSs as described in Section 32 are 1199011198883(1199091 gt 1199092) = 991199011198883(1199091 = 1199092) = 09 1199011198883(1199091 lt 1199092) = 09 1199011198883(1199091 gt 1199093) = 041199011198883(1199091 = 1199093) = 14 1199011198883(1199091 lt 1199093) = 34 1199011198883(1199092 gt 1199093) =

036 1199011198883(1199092 = 1199093) = 036 1199011198883(1199092 lt 1199093) = 3636

Step 2 Aggregate the preference relations using the weightedaverage method 119901(119909119895 gt 119909119896) = sum(119908119894 times 119901119888119894(119909119895 gt 119909119896))119901(119909119895 = 119909119896) = sum(119908119894 times 119901119888i(119909119895 = 119909119896)) and 119901(119909119895 lt 119909119896) =

sum(119908119894 times 119901119888119894(119909119895 lt 119909119896)) where 119908119894 is the weight of criterion 119888119894and sum(119908119894) = 1 In this paper 119908119894 = 13 119894 = 1 2 3 Thusthe final preference relations are (1199091 gt 1199092) = 1521 119901(1199091 =

1199092) = 221 119901(1199091 lt 1199092) = 421 119901(1199091 gt 1199093) = 13119901(1199091 = 1199093) = 112 119901(1199091 lt 1199093) = 712 119901(1199092 gt 1199093) = 13119901(1199092 = 1199093) = 0 119901(1199092 lt 1199093) = 23

Step 3 Rank the alternatives using the nondominance choicedegree method as described in [56] From the results ofStep 2 it can be easily obtained that

119875119878

119863= (

minus

11

21

0

0 minus 0

1

4

1

3

minus

) (16)

Thus NDD1 = min(1 minus 0) (1 minus 14) = 34 NDD2 =

min(1 minus 1121) (1 minus 13) = 1021 and NDD3 = min(1 minus

0) (1 minus 0) = 1 Finally the ranking of alternatives is 1199093 gt

1199091 gt 1199092

6 Mathematical Problems in Engineering

Table 3 Average values of the assessments

1198881

1198882

1198883

1199091

2 45 41199092

25 3 11199093

5 15 5

Table 4 Aggregation results of each alternative

1199091

1199092

1199093

Aggregation result 35 22 38

42 Multicriteria Decision-Making Based onthe Distance Measures of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe average values of HFLTSs for all the alternatives 119909119895 (119895 =

1 2 3) The results are shown in Table 3

Step 2 Aggregate the average values using the weightedaverage method The results are shown in Table 4

Step 3 Rank the alternatives using the distance measuremethod Thus the ranking of alternatives is 1199093

03

≻ 1199091

13

≻ 1199092

43 Results Analysis In [56] the ranking of alternatives is1199091 ≻ 1199093 ≻ 1199092 while both methods in this paper are1199093 ≻ 1199091 ≻ 1199092 Note that the practical decision-makingproblem is quite different from other applications where well-establishedmeasures can be used to quantify the performancefor validation In decision-making usually there is no groundtruth data or quantitativemeasures to assess the performanceof amethod [37]This is why ldquoplausibilityrdquo is used rather thanldquovalidationrdquo Here we analyze the original assessments abouteach criterion of alternatives 1199091 and 1199093 Considering criterion1198881 the original assessments of 1199091 and 1199093 are ldquobetween vland mrdquo and ldquogreater than hrdquo respectively so it is obviously1199093 ≻ 1199091 about criterion 1198881 Considering criterion 1198882 theoriginal assessments of 1199091 and 1199093 are ldquobetween h and vhrdquoand ldquobetween vl and lrdquo respectively so this time 1199091 ≻ 1199093Considering criterion 1198883 the original assessments of 1199091 and1199093 are ldquohrdquo and ldquogreater than hrdquo respectively so 1199093 ≻ 1199091 againSummarily 1199093 ≻ 1199091 occurs twice while 1199091 ≻ 1199093 only onceThus we believe that our result is more plausible

5 Conclusion

The comparison and distance measure of HFLTSs are fun-damentally important in many decision-making problemsunder hesitant fuzzy linguistic environments From an exam-ple we found that there existed two defects when comparingHFLTSs using the previous preference degree method Byanalyzing the definition of an HFLTS a new comparisonmethod based on pairwise comparisons of each linguisticterm in the two HFLTSs has been put forward This compar-ison method does not need the assumption that the valuesin all HFLTSs are arranged in an increasing order and two

HFLTSs have the same length when comparing them Thenwe have defined a distancemeasuremethod betweenHFLTSsbased on pairwise comparisons Further we have proved thatthis distance is equal to the distance of the average values ofHFLTSs which makes the distance measure much simplerFinally two new methods for multicriteria decision-makingin which experts provide their assessments by HFLTSs havebeen proposedThe encouraging results demonstrate that ourmethods in this paper are more reasonable

In the future the application of HFLTSs to groupdecision-making problems will be explored We will alsoinvestigate how to obtain the weights of criteria underhesitant fuzzy linguistic environments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[2] M Mizumoto and K Tanaka ldquoSome properties of fuzzy sets oftype 2rdquo Information andControl vol 31 no 4 pp 312ndash340 1976

[3] J M Mendel and R I B John ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[4] J M Mendel and H Wu ldquoType-2 fuzzistics for symmetricinterval type-2 fuzzy sets part 1 forward problemsrdquo IEEETransactions on Fuzzy Systems vol 14 no 6 pp 781ndash792 2006

[5] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[6] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[7] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[8] K T Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[9] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[10] H Bustince and P Burillo ldquoVague sets are intuitionistic fuzzysetsrdquo Fuzzy Sets and Systems vol 79 no 3 pp 403ndash405 1996

[11] R R Yager ldquoOn the theory of bagsrdquo International Journal ofGeneral Systems vol 13 no 1 pp 23ndash37 1986

[12] S Miyamoto ldquoRemarks on basics of fuzzy sets and fuzzymultisetsrdquo Fuzzy Sets and Systems vol 156 no 3 pp 427ndash4312005

[13] J M Garibaldi M Jaroszewski and S Musikasuwan ldquoNonsta-tionary fuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 16no 4 pp 1072ndash1086 2008

[14] D Li J Han X Shi and M C Chan ldquoKnowledge representa-tion and discovery based on linguistic atomsrdquoKnowledge-BasedSystems vol 10 no 7 pp 431ndash440 1998

[15] D Li C Liu andWGan ldquoAnew cognitivemodel cloudmodelrdquoInternational Journal of Intelligent Systems vol 24 no 3 pp357ndash375 2009

Mathematical Problems in Engineering 7

[16] X Yang L Zeng F Luo and S Wang ldquoCloud hierarchicalanalysisrdquo Journal of Information amp Computational Science vol7 no 12 pp 2468ndash2477 2010

[17] X Yang L Zeng and R Zhang ldquoCloud Delphi methodrdquoInternational Journal of Uncertainty Fuzziness and Knowlege-Based Systems vol 20 no 1 pp 77ndash97 2012

[18] X Yang L Yan and L Zeng ldquoHow to handle uncertaintiesin AHP the Cloud Delphi hierarchical analysisrdquo InformationSciences vol 222 pp 384ndash404 2013

[19] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[20] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju Island Korea August 2009

[21] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 3 pp 199ndash249 1975

[22] L A Zadeh ldquoThe concept of a linguistic variable and its appli-cation to approximate reasoningmdashIIrdquo Information Sciences vol4 pp 301ndash357 1975

[23] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIIIrdquo Information Sciences vol9 no 1 pp 43ndash80 1975

[24] J Ma D Ruan Y Xu and G Zhang ldquoA fuzzy-set approach totreat determinacy and consistency of linguistic terms in multi-criteria decision makingrdquo International Journal of ApproximateReasoning vol 44 no 2 pp 165ndash181 2007

[25] G Bordogna and G Pasi ldquoA fuzzy linguistic approach general-izing boolean information retrieval amodel and its evaluationrdquoJournal of the American Society for Information Science vol 44no 2 pp 70ndash82 1993

[26] G Bordogna and G Pasi ldquoAn ordinal information retrievalmodelrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 9 supplement 1 2001

[27] J Kacprzyk and S Zadrozny ldquoLinguistic database summariesand their protoforms towards natural language based knowl-edge discovery toolsrdquo Information Sciences vol 173 no 4 pp281ndash304 2005

[28] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[29] H Ishibuchi T Nakashima and M Nii Classification andModeling with Linguistic Information Granules AdvancedApproaches to Linguistic Data Mining Springer Berlin Ger-many 2004

[30] R Degani and G Bortolan ldquoThe problem of linguistic approx-imation in clinical decision makingrdquo International Journal ofApproximate Reasoning vol 2 no 2 pp 143ndash162 1988

[31] E Sanchez ldquoTruth-qualification and fuzzy relations in naturallanguages application to medical diagnosisrdquo Fuzzy Sets andSystems vol 84 no 2 pp 155ndash167 1996

[32] H Lee ldquoApplying fuzzy set theory to evaluate the rate ofaggregative risk in software developmentrdquo Fuzzy Sets andSystems vol 79 no 3 pp 323ndash336 1996

[33] L Martınez ldquoSensory evaluation based on linguistic decisionanalysisrdquo International Journal of Approximate Reasoning vol44 no 2 pp 148ndash164 2007

[34] L Martınez J Liu D Ruan and J Yang ldquoDealing with het-erogeneous information in engineering evaluation processesrdquoInformation Sciences vol 177 no 7 pp 1533ndash1542 2007

[35] J Lu Y Zhu X Zeng L Koehl J Ma and G Zhang ldquoA linguis-tic multi-criteria group decision support system for fabric handevaluationrdquo FuzzyOptimization andDecisionMaking vol 8 no4 pp 395ndash413 2009

[36] R de Andres M Espinilla and L Martınez ldquoAn extendedhierarchical linguistic model for managing integral evaluationrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 486ndash500 2010

[37] D Wu and J M Mendel ldquoComputing with words for hierar-chical decision making applied to evaluating a weapon systemrdquoIEEE Transactions on Fuzzy Systems vol 18 no 3 pp 441ndash4602010

[38] F Herrera and E Herrera-Viedma ldquoChoice functions andmechanisms for linguistic preference relationsrdquo European Jour-nal of Operational Research vol 120 no 1 pp 144ndash161 2000

[39] F Herrera E Herrera-Viedma and L Martınez ldquoA fusionapproach for managingmulti-granularity linguistic term sets indecision makingrdquo Fuzzy Sets and Systems vol 114 no 1 pp 43ndash58 2000

[40] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[41] B Arfi ldquoFuzzy decisionmaking in politics a linguistic fuzzy-setapproachrdquo Political Analysis vol 13 no 1 pp 23ndash56 2005

[42] V Huynh and Y Nakamori ldquoA satisfactory-oriented approachto multiexpert decision-making with linguistic assessmentsrdquoIEEE Transactions on Systems Man and Cybernetics B Cyber-netics vol 35 no 2 pp 184ndash196 2005

[43] D Ben-Arieh and Z Chen ldquoLinguistic group decision-makingopinion aggregation and measures of consensusrdquo Fuzzy Opti-mization and Decision Making vol 5 no 4 pp 371ndash386 2006

[44] J L Garcıa-Lapresta ldquoA general class of simple majority deci-sion rules based on linguistic opinionsrdquo Information Sciencesvol 176 no 4 pp 352ndash365 2006

[45] J Wang and J Hao ldquoFuzzy linguistic PERTrdquo IEEE Transactionson Fuzzy Systems vol 15 no 2 pp 133ndash144 2007

[46] T C Wang and Y H Chen ldquoApplying fuzzy linguistic prefer-ence relations to the improvement of consistency of fuzzy AHPrdquoInformation Sciences vol 178 no 19 pp 3755ndash3765 2008

[47] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[48] P D Liu ldquoA novel method for hybridmultiple attribute decisionmakingrdquo Knowledge-Based Systems vol 22 no 5 pp 388ndash3912009

[49] Y Dong W Hong Y Xu and S Yu ldquoSelecting the individualnumerical scale and prioritization method in the analytichierarchy process a 2-Tuple fuzzy linguistic approachrdquo IEEETransactions on Fuzzy Systems vol 19 no 1 pp 13ndash25 2011

[50] L A Zadeh ldquoFuzzy logic = computing with wordsrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 103ndash111 1996

[51] J M Mendel ldquoFuzzy sets for words a new beginningrdquo inProceedings of the IEEE International conference on FuzzySystems pp 37ndash42 St Louis Mo USA May 2003

[52] J MMendel ldquoComputing with words and its relationships withfuzzisticsrdquo Information Sciences vol 177 no 4 pp 988ndash10062007

[53] F Liu and J M Mendel ldquoEncoding words into interval type-2 fuzzy sets using an interval approachrdquo IEEE Transactions onFuzzy Systems vol 16 no 6 pp 1503ndash1521 2008

8 Mathematical Problems in Engineering

[54] S Coupland J M Mendel and D Wu ldquoEnhanced intervalapproach for encoding words into interval type-2 fuzzy sets andconvergence of the word FOUsrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10) pp1ndash8 Barcelona Spain July 2010

[55] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[56] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[57] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[58] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[59] W Wang and X Xin ldquoDistance measure between intuitionisticfuzzy setsrdquo Pattern Recognition Letters vol 26 no 13 pp 2063ndash2069 2005

[60] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[61] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[62] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[63] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[64] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[65] Y Xu and H Wang ldquoDistance measure for linguistic decisionmakingrdquo Systems Engineering Procedia vol 1 pp 450ndash456 2011

[66] Y M Wang J B Yang and D L Xu ldquoA preference aggregationmethod through the estimation of utility intervalsrdquo Computersand Operations Research vol 32 no 8 pp 2027ndash2049 2005

[67] H Liao Z Xu and X Zeng ldquoDistance and similarity measuresfor hesitant fuzzy linguistic term sets and their application inmulti-criteria decision makingrdquo Information Sciences vol 271pp 125ndash142 2014

[68] B Zhu Z Xu andMXia ldquoHesitant fuzzy geometric Bonferronimeansrdquo Information Sciences vol 205 pp 72ndash85 2012

[69] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[70] X Gu Y Wang and B Yang ldquoA method for hesitant fuzzymultiple attribute decision making and its application to riskinvestmentrdquo Journal of Convergence Information Technologyvol 6 no 6 pp 282ndash287 2011

[71] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013

[72] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012

[73] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Example 6 Let 119878 be a linguistic term set 119878 = 1199040

n (nothing) 1199041 vl (very low) 1199042 l (low) 1199043 m (medium)1199044 h (high) 1199045 vh (very high) and 1199046 p (perfect) twodifferent HFLTSs might be 119867

1

119878= 1199041 vl 1199042 l 1199043 m and

1198672

119878= 1199043 m 1199044 h

Definition 7 One defines the number of linguistic terms inthe HFLTS 119867119878 as the cardinality of 119867119878 denoted by |119867119878| InExample 6 |1198671

119878| = 3 and |119867

2

119878| = 2

Definition 8 (see [56]) The lower bound 119867119878minus and upperbound 119867119878+ of the HFLTS 119867119878 are defined as 119867119878minus = min119904119894 |119904119894 isin 119867119878 and 119867119878+ = max119904119894 | 119904119894 isin 119867119878

Definition 9 (see [56]) The envelope of the HFLTS 119867119878env(119867119878) is defined as the linguistic interval [119867119878minus 119867119904+] =

[Ind(119867119878minus) Ind(119867119878+)] where Ind provides the index of thelinguistic term that is Ind(119904119894) = 119894 In Example 6 env(1198671

119878) =

[1199041 1199043] = [1 3] and env(1198672119878) = [1199043 1199044] = [3 4]

Based on the definition of envelope Rodriguez et al [56]compare twoHFLTSs using the comparisonmethod betweentwo numerical intervals introduced by Wang et al [66]

Definition 10 (see [66]) Letting 119860 = [1198861 1198862] and 119861 = [1198871 1198872]

be two intervals the preference degree of119860 over 119861 (or119860 gt 119861)is defined as

119875 (119860 gt 119861) =

max (0 1198862 minus 1198871) minus max (0 1198861 minus 1198872)

(1198862 minus 1198861) + (1198872 minus 1198871)

(3)

and the preference degree of 119861 over 119860 (or 119861 gt 119860) is definedas

119875 (119861 gt 119860) =

max (0 1198872 minus 1198861) minus max (0 1198871 minus 1198862)

(1198862 minus 1198861) + (1198872 minus 1198871)

(4)

Example 11 Let 1198671

119878= 1199041 1199042 1199043 119867

2

119878= 1199043 1199044 and

1198673

119878= 1199045 1199046 be three different HFLTSs on 119878 According to

Definition 9 we have env(1198672119878) = [1 3] env(1198672

119878) = [3 4]

and env(1198673119878) = [5 6] The preference degrees calculated by

Definition 10 (3) and (4) are

119875 (1198671

119878gt 1198672

119878) = 0 119875 (119867

2

119878gt 1198671

119878) = 1

119875 (1198671

119878gt 1198673

119878) = 0 119875 (119867

3

119878gt 1198671

119878) = 1

119875 (1198672

119878gt 1198673

119878) = 0 119875 (119867

3

119878gt 1198672

119878) = 1

(5)

From Example 11 mentioned above it can be observedthat when we compare two HFLTSs using the preferencedegree method there exist two defects as follows

(1) The result 119875(1198672

119878gt 119867

1

119878) = 1 indicates that 119867

2

119878

is absolutely superior to 1198671

119878 In fact both 119867

1

119878and 119867

2

119878

contain the linguistic term 1199043 It means that the value of alinguistic variable may be equal in these two cases Thus itis unreasonable to say that 1198672

119878is absolutely superior to 119867

1

119878

(2) The result 119875(1198673

119878gt 1198671

119878) = 119875(119867

2

119878gt 1198671

119878) = 1 meaning

that when comparedwith1198671

119878 the twoHFLTSs1198672

119878and119867

3

119878are

identical In fact 1198673119878is more superior to 119867

1

119878compared to 119867

2

119878

to 1198671

119878 Thus using the preference degree method to compare

HFLTSs may result in losing some important informationBased on the analysis mentioned above we think that it

is not suitable to compare discrete linguistic terms in HFLTSsusing the comparison method for continuous numericalintervals By the definition of an HFLTS we know thatevery linguistic term in it is a possible value of the linguisticinformation And noting that the twoHFLTSs for comparingmay have different lengths So when comparing twoHFLTSsit needs pairwise comparisons of each linguistic term in them

3 Comparison and DistanceMeasure of HFLTSs

31 Distance between Two Single Linguistic Terms Let 119904119894 119904119895 isin119878 be two linguistic terms Xu [64] defined the deviationmeasure between 119904119894 and 119904119895 as follows

119889 (119904119894 119904119895) =

1003816100381610038161003816119894 minus 119895

1003816100381610038161003816

119879

(6)

where 119879 is the cardinality of 119878 that is 119879 = |119878|If only one preestablished linguistic term set 119878 is used in

a decision-making model we can simply consider [49 65]119889(119904119894 119904119895) = |119894 minus 119895| = 119904|119894minus119895|

Definition 12 Letting 119904119894 119904119895 isin 119878 be two single linguistic termsthen we call

119889 (119904119894 119904119895) = 119904119894 minus 119904119895 = 119894 minus 119895 (7)

the distance between 119904119894 and 119904119895

The distance measure between 119904119894 and 119904119895 has a definitephysical implication and reflects the relative position anddistance between 119904119894 and 119904119895 If 119889(119904119894 119904119895) = 0 then 119904119894 = 119904119895 If119889(119904119894 119904119895) gt 0 then 119904119894 gt 119904119895 If 119889(119904119894 119904119895) lt 0 then 119904119894 lt 119904119895

Theorem13 Letting 119904119894 119904119895 119904119896 isin 119878 be three linguistic terms then(1) 119889(119904119894 119904119895) = minus119889(119904119895 119904119894)(2) (|119878| minus 1) le 119889(119904119894 119904119895) le (|119878| minus 1)(3) 119889(119904119894 119904119896) = 119889(119904119894 119904119895) + 119889(119904119895 119904119896)

Proof They are straightforward and thus omitted

32 Comparison of HFLTSs The comparison of HFLTSs isnecessary in many problems such as ranking and selectionHowever anHFLTS is a linguistic term subset which containsseveral linguistic terms and the comparison among HFLTSsis not simple Here a new comparison method of HFLTSswhich is based on pairwise comparisons of each linguisticterm in the two HFLTSs is put forward

Definition 14 Letting 1198671

119878and 119867

2

119878be two HFLTSs on 119878 then

one defines the pairwise comparison matrix between1198671

119878and

1198672

119878as follows

119862 (1198671

119878 1198672

119878) = [119889 (119904119894 119904119895)]|1198671

119878|times|1198672119878| 119904119894 isin 119867

1

119878 119904119895 isin 119867

2

119878 (8)

4 Mathematical Problems in Engineering

Remark 15 The number of linguistic terms in the twoHFLTSs 1198671

119878and 119867

2

119878 may be unequal that is |1198671

119878| = |119867

2

119878|

To deal with such situations usually it is necessary to extendthe shorter one by adding the stated value several times init [62 63] while our pairwise comparison method does notrequire this step

Remark 16 From Definition 14 we have [119862(1198671

119878 1198672

119878)] =

minus[119862(1198672

119878 1198671

119878)]

119879 where 119879 is the transpose operator of matrix

Example 17 Let 1198671119878

= 1199041 1199042 1199043 and 1198672

119878= 1199042 1199043 1199044 1199045 be

twoHFLTSs on 119878 According toDefinition 14 the comparisonmatrix 119862 between 119867

1

119878and 119867

2

119878is

1199041

1199042

1199043

1199042 1199043 1199044 1199045

[

[

minus1

0

1

minus2

minus1

0

minus3

minus2

minus1

minus4

minus3

minus2

]

]

(9)

Definition 18 Letting 119862 = 119862(1198671

119878 1198672

119878) be the pairwise

comparison matrix between 1198671

119878and 119867

2

119878 the preference

relations of 1198671119878and 119867

2

119878are defined as follows

119901 (1198671

119878gt 1198672

119878) =

10038161003816100381610038161003816sum119862119898119899gt0

119862119898119899

10038161003816100381610038161003816

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

119901 (1198671

119878= 1198672

119878) =

119862119898119899 = 0

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

119901 (1198671

119878lt 1198672

119878) =

10038161003816100381610038161003816sum119862119898119899lt0

119862119898119899

10038161003816100381610038161003816

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

(10)

It is obvious that 119901(1198671

119878gt 1198672

119878) + 119901(119867

1

119878= 1198672

119878) + 119901(119867

1

119878lt

1198672

119878) = 1 We say that 1198671

119878is superior to 119867

2

119878with the degree of

119901(1198671

119878gt 1198672

119878) denoted by 119867

1

119878≻119901(1198671

119878gt1198672

119878)1198672

119878 1198671119878is equal to 119867

2

119878

with the degree of 119901(1198671

119878= 1198672

119878) denoted by 119867

1

119878sim119901(1198671

119878=1198672

119878)1198672

119878

and 1198671

119878is inferior to 119867

2

119878with the degree of 119901(119867

1

119878lt 1198672

119878)

denoted by 1198671

119878≺119901(1198671

119878lt1198672

119878)1198672

119878

Considering Example 17 by Definition 18 (10) the pref-erence relations of 119867

1

119878and 119867

2

119878were calculated as 119901(119867

1

119878gt

1198672

119878) = 122 119901(119867

1

119878= 1198672

119878) = 222 and 119901(119867

1

119878lt 1198672

119878) = 1922

Thus the comparison results are1198671

119878≻122

1198672

1198781198671119878sim222

1198672

119878 and

1198671

119878≺1922

1198672

119878

33 Distance Measure of HFLTSs

Definition 19 Letting 119862 = 119862(1198671

119878 1198672

119878) be the pairwise

comparisonmatrix between1198671

119878and119867

2

119878 the distance between

1198671

119878and 119867

2

119878is defined as the average value of the pairwise

comparison matrix

119889 (1198671

119878 1198672

119878) =

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

|1198671

119878|

sum

119898=1

|1198672

119878|

sum

119899=1

119862119898119899(11)

Considering Example 17 one has 119889(1198671

119878 1198672

119878) = (minus18)

(3 times 4) = minus15

To preserve all the given information the discrete linguis-tic term set 119878 is extended to a continuous term set 119878 = 119904120572 |

120572 isin [minus119902 119902] where 119902 is a sufficiently large positive number If119904120572 isin 119878 then we call 119904120572 an original linguistic term otherwisewe call 119904120572 a virtual linguistic term

Remark 20 In general the decision-maker uses the originallinguistic terms to express hisher qualitative opinions andthe virtual linguistic terms can only appear in operations

Definition 21 The average value of anHFLTS119867119878 is defined as

Aver (119867119878) =

1

1003816100381610038161003816119867119878

1003816100381610038161003816

sum

119904119894isin119867119878

119904119894 =

1

1003816100381610038161003816119867119878

1003816100381610038161003816

sum

119904119894isin119867119878

Ind (119904119894) (12)

This definition is similar to the score function of an HFEDefinition 3

Considering Example 17 we have Aver(1198671119878) = 119904(1+2+3)3 =

1199042 = 2 and Aver(1198672119878) = 119904(2+3+4+5)4 = 11990435 =35

Theorem 22 Letting 119867119878 be an HFLTS on 119878 then0 le 119867119904minus le Aver (119867119878) le 119867119904+ le (|119878| minus 1) (13)

Proof It is straightforward and thus omitted

Theorem 23 Letting 1198671

119878and 119867

2

119878be two HFLTSs on 119878 the

distance between 1198671

119878and 119867

2

119878defined by the average value of

their pairwise comparison matrix is equal to the distance of thetwo average values of1198671

119878and119867

2

119878 that is the distance between

1198671

119878and 119867

2

119878can be easily obtained by

119889 (1198671

119878 1198672

119878) = Aver (1198671

119878) minus Aver (1198672

119878) (14)

Proof From Definitions 19 and 14 we have

119889 (1198671

119878 1198672

119878) =

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

|1198671

119878|

sum

119898=1

|1198672

119878|

sum

119899=1

119862119898119899

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119889 (119904119894 119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

(119904119894 minus 119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

( sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119904119894 minus sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

(

100381610038161003816100381610038161198672

119878

10038161003816100381610038161003816times sum

119904119894isin1198671119878

119904119894 minus

100381610038161003816100381610038161198671

119878

10038161003816100381610038161003816times sum

119904119895isin1198672119878

119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

119904119894 minus

1

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119895isin1198672119878

119904119895

= Aver (1198671119878) minus Aver (1198672

119878)

(15)

which completes the proof of Theorem 23

Mathematical Problems in Engineering 5

Considering Example 17 we have 119889(1198671

119878 1198672

119878) =

Aver(1198671119878) minus Aver(1198672

119878) = 2 minus 35 = minus15

By Theorem 23 we can easily obtain the following corol-lary

Corollary 24 Letting 1198671

119878 1198672119878and 119867

3

119878be three HFLTSs on S

then

(1) (|119878| minus 1) le 119889(1198671

119878 1198672

119878) le (|119878| minus 1)

(2) 119889(1198671

119878 1198672

119878) = minus119889(119867

2

119878 1198671

119878)

(3) 119889(1198671

119878 1198673

119878) = 119889(119867

1

119878 1198672

119878) + 119889(119867

2

119878 1198673

119878)

Proof They are straightforward and thus omitted

If 119889(1198671

119878 1198672

119878) gt 0 (or Aver(1198671

119878) gt Aver(1198672

119878)) then we

say that 1198671119878is superior to 119867

2

119878with the distance of 119889(1198671

119878 1198672

119878)

denoted by 1198671

119878

119889(1198671

1198781198672

119878)

≻ 1198672

119878 if 119889(1198671

119878 1198672

119878) = 0 (or Aver(1198671

119878) =

Aver(1198672119878)) then we say that 1198671

119878is indifferent to 119867

2

119878 denoted

by 1198671

119878sim 1198672

119878 if 119889(1198671

119878 1198672

119878) lt 0 (or Aver(1198671

119878) lt Aver(1198672

119878))

then we say that 1198671

119878is inferior to 119867

2

119878with the distance of

119889(1198671

119878 1198672

119878) denoted by 119867

1

119878

119889(1198671

1198781198672

119878)

≺ 1198672

119878

4 Multicriteria Decision-Making ModelsBased on Comparisons and DistanceMeasures of HFLTSs

In this section two new methods are presented for rankingand choice from a set of alternatives in the framework ofmulticriteria decision-making using linguistic informationOne is based on the comparisons and preference relationsof HFLTSs and the other is based on the distance measureof HFLTSs We adopt Example 5 in [56] (Example 25 in ourpaper) to illustrate the detailed processes of the twomethods

Example 25 ([see [56]) Let 119883 = 1199091 1199092 1199093 be a set ofalternatives 119862 = 1198881 1198882 1198883 a set of criteria defined for eachalternative and 119878 = 1199040 n (nothing) 1199041 vl (very low)1199042 l (low) 1199043 m (medium) 1199044 h (high) 1199045

vh (very high) 1199046 p (perfect) the linguistic term set that isused to generate the linguistic expressions The assessmentsthat are provided in such a problem are shown in Table 1 andthey are transformed into HFLTSs as shown in Table 2

41 Multicriteria Decision-Making Based onthe Comparisons of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe preference degrees between all the alternatives 119909119895 (119895 =

1 2 3)Considering criterion 1198881 119867

1199091

119878= 1199041 1199042 1199043 119867

1199092

119878=

1199042 1199043 and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees

about criterion 1198881 calculated using the comparisonmethod ofHFLTSs as described in Section 32 are 1199011198881(1199091 gt 1199092) = 171199011198881(1199091 = 1199092) = 27 1199011198881(1199091 lt 1199092) = 47 1199011198881(1199091 gt 1199093) =

027 1199011198881(1199091 = 1199093) = 027 1199011198881(1199091 lt 1199093) = 2727 1199011198881(1199092 gt

1199093) = 015 1199011198881(1199092 = 1199093) = 015 1199011198881(1199092 lt 1199093) = 1515

Table 1 Assessments that are provided for the decision problem

1198881

1198882

1198883

1199091

Between vl and m Between h and vh h1199092

Between l and m m Lower than l1199093

Greater than h Between vl and l Greater than h

Table 2 Assessments transformed into HFLTSs

1198881

1198882

1198883

1199091

1199041 1199042 1199043 119904

4 1199045 119904

4

1199092

1199042 1199043 119904

3 119904

0 1199041 1199042

1199093

1199044 1199045 1199046 119904

1 1199042 119904

4 1199045 1199046

Considering criterion 1198882 1198671199091

119878= 1199044 1199045 119867

1199092

119878= 1199043

and 1198671199093

119878= 1199041 1199042 so the preference degrees about criterion

1198882 calculated using the comparison method of HFLTSs asdescribed in Section 32 are 1199011198882(1199091 gt 1199092) = 33 1199011198882(1199091 =

1199092) = 03 1199011198882(1199091 lt 1199092) = 03 1199011198882(1199091 gt 1199093) = 12121199011198882(1199091 = 1199093) = 012 1199011198882(1199091 lt 1199093) = 012 1199011198882(1199092 gt 1199093) =

33 1199011198882(1199092 = 1199093) = 03 1199011198882(1199092 lt 1199093) = 03Considering criterion 1198883 119867

1199091

119878= 1199044 119867

1199092

119878= 1199040 1199041 1199042

and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees about

criterion 1198883 calculated using the comparison method ofHFLTSs as described in Section 32 are 1199011198883(1199091 gt 1199092) = 991199011198883(1199091 = 1199092) = 09 1199011198883(1199091 lt 1199092) = 09 1199011198883(1199091 gt 1199093) = 041199011198883(1199091 = 1199093) = 14 1199011198883(1199091 lt 1199093) = 34 1199011198883(1199092 gt 1199093) =

036 1199011198883(1199092 = 1199093) = 036 1199011198883(1199092 lt 1199093) = 3636

Step 2 Aggregate the preference relations using the weightedaverage method 119901(119909119895 gt 119909119896) = sum(119908119894 times 119901119888119894(119909119895 gt 119909119896))119901(119909119895 = 119909119896) = sum(119908119894 times 119901119888i(119909119895 = 119909119896)) and 119901(119909119895 lt 119909119896) =

sum(119908119894 times 119901119888119894(119909119895 lt 119909119896)) where 119908119894 is the weight of criterion 119888119894and sum(119908119894) = 1 In this paper 119908119894 = 13 119894 = 1 2 3 Thusthe final preference relations are (1199091 gt 1199092) = 1521 119901(1199091 =

1199092) = 221 119901(1199091 lt 1199092) = 421 119901(1199091 gt 1199093) = 13119901(1199091 = 1199093) = 112 119901(1199091 lt 1199093) = 712 119901(1199092 gt 1199093) = 13119901(1199092 = 1199093) = 0 119901(1199092 lt 1199093) = 23

Step 3 Rank the alternatives using the nondominance choicedegree method as described in [56] From the results ofStep 2 it can be easily obtained that

119875119878

119863= (

minus

11

21

0

0 minus 0

1

4

1

3

minus

) (16)

Thus NDD1 = min(1 minus 0) (1 minus 14) = 34 NDD2 =

min(1 minus 1121) (1 minus 13) = 1021 and NDD3 = min(1 minus

0) (1 minus 0) = 1 Finally the ranking of alternatives is 1199093 gt

1199091 gt 1199092

6 Mathematical Problems in Engineering

Table 3 Average values of the assessments

1198881

1198882

1198883

1199091

2 45 41199092

25 3 11199093

5 15 5

Table 4 Aggregation results of each alternative

1199091

1199092

1199093

Aggregation result 35 22 38

42 Multicriteria Decision-Making Based onthe Distance Measures of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe average values of HFLTSs for all the alternatives 119909119895 (119895 =

1 2 3) The results are shown in Table 3

Step 2 Aggregate the average values using the weightedaverage method The results are shown in Table 4

Step 3 Rank the alternatives using the distance measuremethod Thus the ranking of alternatives is 1199093

03

≻ 1199091

13

≻ 1199092

43 Results Analysis In [56] the ranking of alternatives is1199091 ≻ 1199093 ≻ 1199092 while both methods in this paper are1199093 ≻ 1199091 ≻ 1199092 Note that the practical decision-makingproblem is quite different from other applications where well-establishedmeasures can be used to quantify the performancefor validation In decision-making usually there is no groundtruth data or quantitativemeasures to assess the performanceof amethod [37]This is why ldquoplausibilityrdquo is used rather thanldquovalidationrdquo Here we analyze the original assessments abouteach criterion of alternatives 1199091 and 1199093 Considering criterion1198881 the original assessments of 1199091 and 1199093 are ldquobetween vland mrdquo and ldquogreater than hrdquo respectively so it is obviously1199093 ≻ 1199091 about criterion 1198881 Considering criterion 1198882 theoriginal assessments of 1199091 and 1199093 are ldquobetween h and vhrdquoand ldquobetween vl and lrdquo respectively so this time 1199091 ≻ 1199093Considering criterion 1198883 the original assessments of 1199091 and1199093 are ldquohrdquo and ldquogreater than hrdquo respectively so 1199093 ≻ 1199091 againSummarily 1199093 ≻ 1199091 occurs twice while 1199091 ≻ 1199093 only onceThus we believe that our result is more plausible

5 Conclusion

The comparison and distance measure of HFLTSs are fun-damentally important in many decision-making problemsunder hesitant fuzzy linguistic environments From an exam-ple we found that there existed two defects when comparingHFLTSs using the previous preference degree method Byanalyzing the definition of an HFLTS a new comparisonmethod based on pairwise comparisons of each linguisticterm in the two HFLTSs has been put forward This compar-ison method does not need the assumption that the valuesin all HFLTSs are arranged in an increasing order and two

HFLTSs have the same length when comparing them Thenwe have defined a distancemeasuremethod betweenHFLTSsbased on pairwise comparisons Further we have proved thatthis distance is equal to the distance of the average values ofHFLTSs which makes the distance measure much simplerFinally two new methods for multicriteria decision-makingin which experts provide their assessments by HFLTSs havebeen proposedThe encouraging results demonstrate that ourmethods in this paper are more reasonable

In the future the application of HFLTSs to groupdecision-making problems will be explored We will alsoinvestigate how to obtain the weights of criteria underhesitant fuzzy linguistic environments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

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[2] M Mizumoto and K Tanaka ldquoSome properties of fuzzy sets oftype 2rdquo Information andControl vol 31 no 4 pp 312ndash340 1976

[3] J M Mendel and R I B John ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[4] J M Mendel and H Wu ldquoType-2 fuzzistics for symmetricinterval type-2 fuzzy sets part 1 forward problemsrdquo IEEETransactions on Fuzzy Systems vol 14 no 6 pp 781ndash792 2006

[5] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[6] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[7] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[8] K T Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[9] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[10] H Bustince and P Burillo ldquoVague sets are intuitionistic fuzzysetsrdquo Fuzzy Sets and Systems vol 79 no 3 pp 403ndash405 1996

[11] R R Yager ldquoOn the theory of bagsrdquo International Journal ofGeneral Systems vol 13 no 1 pp 23ndash37 1986

[12] S Miyamoto ldquoRemarks on basics of fuzzy sets and fuzzymultisetsrdquo Fuzzy Sets and Systems vol 156 no 3 pp 427ndash4312005

[13] J M Garibaldi M Jaroszewski and S Musikasuwan ldquoNonsta-tionary fuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 16no 4 pp 1072ndash1086 2008

[14] D Li J Han X Shi and M C Chan ldquoKnowledge representa-tion and discovery based on linguistic atomsrdquoKnowledge-BasedSystems vol 10 no 7 pp 431ndash440 1998

[15] D Li C Liu andWGan ldquoAnew cognitivemodel cloudmodelrdquoInternational Journal of Intelligent Systems vol 24 no 3 pp357ndash375 2009

Mathematical Problems in Engineering 7

[16] X Yang L Zeng F Luo and S Wang ldquoCloud hierarchicalanalysisrdquo Journal of Information amp Computational Science vol7 no 12 pp 2468ndash2477 2010

[17] X Yang L Zeng and R Zhang ldquoCloud Delphi methodrdquoInternational Journal of Uncertainty Fuzziness and Knowlege-Based Systems vol 20 no 1 pp 77ndash97 2012

[18] X Yang L Yan and L Zeng ldquoHow to handle uncertaintiesin AHP the Cloud Delphi hierarchical analysisrdquo InformationSciences vol 222 pp 384ndash404 2013

[19] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[20] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju Island Korea August 2009

[21] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 3 pp 199ndash249 1975

[22] L A Zadeh ldquoThe concept of a linguistic variable and its appli-cation to approximate reasoningmdashIIrdquo Information Sciences vol4 pp 301ndash357 1975

[23] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIIIrdquo Information Sciences vol9 no 1 pp 43ndash80 1975

[24] J Ma D Ruan Y Xu and G Zhang ldquoA fuzzy-set approach totreat determinacy and consistency of linguistic terms in multi-criteria decision makingrdquo International Journal of ApproximateReasoning vol 44 no 2 pp 165ndash181 2007

[25] G Bordogna and G Pasi ldquoA fuzzy linguistic approach general-izing boolean information retrieval amodel and its evaluationrdquoJournal of the American Society for Information Science vol 44no 2 pp 70ndash82 1993

[26] G Bordogna and G Pasi ldquoAn ordinal information retrievalmodelrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 9 supplement 1 2001

[27] J Kacprzyk and S Zadrozny ldquoLinguistic database summariesand their protoforms towards natural language based knowl-edge discovery toolsrdquo Information Sciences vol 173 no 4 pp281ndash304 2005

[28] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[29] H Ishibuchi T Nakashima and M Nii Classification andModeling with Linguistic Information Granules AdvancedApproaches to Linguistic Data Mining Springer Berlin Ger-many 2004

[30] R Degani and G Bortolan ldquoThe problem of linguistic approx-imation in clinical decision makingrdquo International Journal ofApproximate Reasoning vol 2 no 2 pp 143ndash162 1988

[31] E Sanchez ldquoTruth-qualification and fuzzy relations in naturallanguages application to medical diagnosisrdquo Fuzzy Sets andSystems vol 84 no 2 pp 155ndash167 1996

[32] H Lee ldquoApplying fuzzy set theory to evaluate the rate ofaggregative risk in software developmentrdquo Fuzzy Sets andSystems vol 79 no 3 pp 323ndash336 1996

[33] L Martınez ldquoSensory evaluation based on linguistic decisionanalysisrdquo International Journal of Approximate Reasoning vol44 no 2 pp 148ndash164 2007

[34] L Martınez J Liu D Ruan and J Yang ldquoDealing with het-erogeneous information in engineering evaluation processesrdquoInformation Sciences vol 177 no 7 pp 1533ndash1542 2007

[35] J Lu Y Zhu X Zeng L Koehl J Ma and G Zhang ldquoA linguis-tic multi-criteria group decision support system for fabric handevaluationrdquo FuzzyOptimization andDecisionMaking vol 8 no4 pp 395ndash413 2009

[36] R de Andres M Espinilla and L Martınez ldquoAn extendedhierarchical linguistic model for managing integral evaluationrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 486ndash500 2010

[37] D Wu and J M Mendel ldquoComputing with words for hierar-chical decision making applied to evaluating a weapon systemrdquoIEEE Transactions on Fuzzy Systems vol 18 no 3 pp 441ndash4602010

[38] F Herrera and E Herrera-Viedma ldquoChoice functions andmechanisms for linguistic preference relationsrdquo European Jour-nal of Operational Research vol 120 no 1 pp 144ndash161 2000

[39] F Herrera E Herrera-Viedma and L Martınez ldquoA fusionapproach for managingmulti-granularity linguistic term sets indecision makingrdquo Fuzzy Sets and Systems vol 114 no 1 pp 43ndash58 2000

[40] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[41] B Arfi ldquoFuzzy decisionmaking in politics a linguistic fuzzy-setapproachrdquo Political Analysis vol 13 no 1 pp 23ndash56 2005

[42] V Huynh and Y Nakamori ldquoA satisfactory-oriented approachto multiexpert decision-making with linguistic assessmentsrdquoIEEE Transactions on Systems Man and Cybernetics B Cyber-netics vol 35 no 2 pp 184ndash196 2005

[43] D Ben-Arieh and Z Chen ldquoLinguistic group decision-makingopinion aggregation and measures of consensusrdquo Fuzzy Opti-mization and Decision Making vol 5 no 4 pp 371ndash386 2006

[44] J L Garcıa-Lapresta ldquoA general class of simple majority deci-sion rules based on linguistic opinionsrdquo Information Sciencesvol 176 no 4 pp 352ndash365 2006

[45] J Wang and J Hao ldquoFuzzy linguistic PERTrdquo IEEE Transactionson Fuzzy Systems vol 15 no 2 pp 133ndash144 2007

[46] T C Wang and Y H Chen ldquoApplying fuzzy linguistic prefer-ence relations to the improvement of consistency of fuzzy AHPrdquoInformation Sciences vol 178 no 19 pp 3755ndash3765 2008

[47] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[48] P D Liu ldquoA novel method for hybridmultiple attribute decisionmakingrdquo Knowledge-Based Systems vol 22 no 5 pp 388ndash3912009

[49] Y Dong W Hong Y Xu and S Yu ldquoSelecting the individualnumerical scale and prioritization method in the analytichierarchy process a 2-Tuple fuzzy linguistic approachrdquo IEEETransactions on Fuzzy Systems vol 19 no 1 pp 13ndash25 2011

[50] L A Zadeh ldquoFuzzy logic = computing with wordsrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 103ndash111 1996

[51] J M Mendel ldquoFuzzy sets for words a new beginningrdquo inProceedings of the IEEE International conference on FuzzySystems pp 37ndash42 St Louis Mo USA May 2003

[52] J MMendel ldquoComputing with words and its relationships withfuzzisticsrdquo Information Sciences vol 177 no 4 pp 988ndash10062007

[53] F Liu and J M Mendel ldquoEncoding words into interval type-2 fuzzy sets using an interval approachrdquo IEEE Transactions onFuzzy Systems vol 16 no 6 pp 1503ndash1521 2008

8 Mathematical Problems in Engineering

[54] S Coupland J M Mendel and D Wu ldquoEnhanced intervalapproach for encoding words into interval type-2 fuzzy sets andconvergence of the word FOUsrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10) pp1ndash8 Barcelona Spain July 2010

[55] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[56] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[57] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[58] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[59] W Wang and X Xin ldquoDistance measure between intuitionisticfuzzy setsrdquo Pattern Recognition Letters vol 26 no 13 pp 2063ndash2069 2005

[60] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[61] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[62] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[63] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[64] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[65] Y Xu and H Wang ldquoDistance measure for linguistic decisionmakingrdquo Systems Engineering Procedia vol 1 pp 450ndash456 2011

[66] Y M Wang J B Yang and D L Xu ldquoA preference aggregationmethod through the estimation of utility intervalsrdquo Computersand Operations Research vol 32 no 8 pp 2027ndash2049 2005

[67] H Liao Z Xu and X Zeng ldquoDistance and similarity measuresfor hesitant fuzzy linguistic term sets and their application inmulti-criteria decision makingrdquo Information Sciences vol 271pp 125ndash142 2014

[68] B Zhu Z Xu andMXia ldquoHesitant fuzzy geometric Bonferronimeansrdquo Information Sciences vol 205 pp 72ndash85 2012

[69] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[70] X Gu Y Wang and B Yang ldquoA method for hesitant fuzzymultiple attribute decision making and its application to riskinvestmentrdquo Journal of Convergence Information Technologyvol 6 no 6 pp 282ndash287 2011

[71] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013

[72] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012

[73] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

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4 Mathematical Problems in Engineering

Remark 15 The number of linguistic terms in the twoHFLTSs 1198671

119878and 119867

2

119878 may be unequal that is |1198671

119878| = |119867

2

119878|

To deal with such situations usually it is necessary to extendthe shorter one by adding the stated value several times init [62 63] while our pairwise comparison method does notrequire this step

Remark 16 From Definition 14 we have [119862(1198671

119878 1198672

119878)] =

minus[119862(1198672

119878 1198671

119878)]

119879 where 119879 is the transpose operator of matrix

Example 17 Let 1198671119878

= 1199041 1199042 1199043 and 1198672

119878= 1199042 1199043 1199044 1199045 be

twoHFLTSs on 119878 According toDefinition 14 the comparisonmatrix 119862 between 119867

1

119878and 119867

2

119878is

1199041

1199042

1199043

1199042 1199043 1199044 1199045

[

[

minus1

0

1

minus2

minus1

0

minus3

minus2

minus1

minus4

minus3

minus2

]

]

(9)

Definition 18 Letting 119862 = 119862(1198671

119878 1198672

119878) be the pairwise

comparison matrix between 1198671

119878and 119867

2

119878 the preference

relations of 1198671119878and 119867

2

119878are defined as follows

119901 (1198671

119878gt 1198672

119878) =

10038161003816100381610038161003816sum119862119898119899gt0

119862119898119899

10038161003816100381610038161003816

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

119901 (1198671

119878= 1198672

119878) =

119862119898119899 = 0

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

119901 (1198671

119878lt 1198672

119878) =

10038161003816100381610038161003816sum119862119898119899lt0

119862119898119899

10038161003816100381610038161003816

119862119898119899 = 0 + sum1003816100381610038161003816119862119898119899

1003816100381610038161003816

(10)

It is obvious that 119901(1198671

119878gt 1198672

119878) + 119901(119867

1

119878= 1198672

119878) + 119901(119867

1

119878lt

1198672

119878) = 1 We say that 1198671

119878is superior to 119867

2

119878with the degree of

119901(1198671

119878gt 1198672

119878) denoted by 119867

1

119878≻119901(1198671

119878gt1198672

119878)1198672

119878 1198671119878is equal to 119867

2

119878

with the degree of 119901(1198671

119878= 1198672

119878) denoted by 119867

1

119878sim119901(1198671

119878=1198672

119878)1198672

119878

and 1198671

119878is inferior to 119867

2

119878with the degree of 119901(119867

1

119878lt 1198672

119878)

denoted by 1198671

119878≺119901(1198671

119878lt1198672

119878)1198672

119878

Considering Example 17 by Definition 18 (10) the pref-erence relations of 119867

1

119878and 119867

2

119878were calculated as 119901(119867

1

119878gt

1198672

119878) = 122 119901(119867

1

119878= 1198672

119878) = 222 and 119901(119867

1

119878lt 1198672

119878) = 1922

Thus the comparison results are1198671

119878≻122

1198672

1198781198671119878sim222

1198672

119878 and

1198671

119878≺1922

1198672

119878

33 Distance Measure of HFLTSs

Definition 19 Letting 119862 = 119862(1198671

119878 1198672

119878) be the pairwise

comparisonmatrix between1198671

119878and119867

2

119878 the distance between

1198671

119878and 119867

2

119878is defined as the average value of the pairwise

comparison matrix

119889 (1198671

119878 1198672

119878) =

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

|1198671

119878|

sum

119898=1

|1198672

119878|

sum

119899=1

119862119898119899(11)

Considering Example 17 one has 119889(1198671

119878 1198672

119878) = (minus18)

(3 times 4) = minus15

To preserve all the given information the discrete linguis-tic term set 119878 is extended to a continuous term set 119878 = 119904120572 |

120572 isin [minus119902 119902] where 119902 is a sufficiently large positive number If119904120572 isin 119878 then we call 119904120572 an original linguistic term otherwisewe call 119904120572 a virtual linguistic term

Remark 20 In general the decision-maker uses the originallinguistic terms to express hisher qualitative opinions andthe virtual linguistic terms can only appear in operations

Definition 21 The average value of anHFLTS119867119878 is defined as

Aver (119867119878) =

1

1003816100381610038161003816119867119878

1003816100381610038161003816

sum

119904119894isin119867119878

119904119894 =

1

1003816100381610038161003816119867119878

1003816100381610038161003816

sum

119904119894isin119867119878

Ind (119904119894) (12)

This definition is similar to the score function of an HFEDefinition 3

Considering Example 17 we have Aver(1198671119878) = 119904(1+2+3)3 =

1199042 = 2 and Aver(1198672119878) = 119904(2+3+4+5)4 = 11990435 =35

Theorem 22 Letting 119867119878 be an HFLTS on 119878 then0 le 119867119904minus le Aver (119867119878) le 119867119904+ le (|119878| minus 1) (13)

Proof It is straightforward and thus omitted

Theorem 23 Letting 1198671

119878and 119867

2

119878be two HFLTSs on 119878 the

distance between 1198671

119878and 119867

2

119878defined by the average value of

their pairwise comparison matrix is equal to the distance of thetwo average values of1198671

119878and119867

2

119878 that is the distance between

1198671

119878and 119867

2

119878can be easily obtained by

119889 (1198671

119878 1198672

119878) = Aver (1198671

119878) minus Aver (1198672

119878) (14)

Proof From Definitions 19 and 14 we have

119889 (1198671

119878 1198672

119878) =

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

|1198671

119878|

sum

119898=1

|1198672

119878|

sum

119899=1

119862119898119899

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119889 (119904119894 119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

(119904119894 minus 119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

( sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119904119894 minus sum

119904119894isin1198671119878

sum

119904119895isin1198672119878

119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816times

10038161003816100381610038161198672

119878

1003816100381610038161003816

(

100381610038161003816100381610038161198672

119878

10038161003816100381610038161003816times sum

119904119894isin1198671119878

119904119894 minus

100381610038161003816100381610038161198671

119878

10038161003816100381610038161003816times sum

119904119895isin1198672119878

119904119895)

=

1

10038161003816100381610038161198671

119878

1003816100381610038161003816

sum

119904119894isin1198671119878

119904119894 minus

1

10038161003816100381610038161198672

119878

1003816100381610038161003816

sum

119904119895isin1198672119878

119904119895

= Aver (1198671119878) minus Aver (1198672

119878)

(15)

which completes the proof of Theorem 23

Mathematical Problems in Engineering 5

Considering Example 17 we have 119889(1198671

119878 1198672

119878) =

Aver(1198671119878) minus Aver(1198672

119878) = 2 minus 35 = minus15

By Theorem 23 we can easily obtain the following corol-lary

Corollary 24 Letting 1198671

119878 1198672119878and 119867

3

119878be three HFLTSs on S

then

(1) (|119878| minus 1) le 119889(1198671

119878 1198672

119878) le (|119878| minus 1)

(2) 119889(1198671

119878 1198672

119878) = minus119889(119867

2

119878 1198671

119878)

(3) 119889(1198671

119878 1198673

119878) = 119889(119867

1

119878 1198672

119878) + 119889(119867

2

119878 1198673

119878)

Proof They are straightforward and thus omitted

If 119889(1198671

119878 1198672

119878) gt 0 (or Aver(1198671

119878) gt Aver(1198672

119878)) then we

say that 1198671119878is superior to 119867

2

119878with the distance of 119889(1198671

119878 1198672

119878)

denoted by 1198671

119878

119889(1198671

1198781198672

119878)

≻ 1198672

119878 if 119889(1198671

119878 1198672

119878) = 0 (or Aver(1198671

119878) =

Aver(1198672119878)) then we say that 1198671

119878is indifferent to 119867

2

119878 denoted

by 1198671

119878sim 1198672

119878 if 119889(1198671

119878 1198672

119878) lt 0 (or Aver(1198671

119878) lt Aver(1198672

119878))

then we say that 1198671

119878is inferior to 119867

2

119878with the distance of

119889(1198671

119878 1198672

119878) denoted by 119867

1

119878

119889(1198671

1198781198672

119878)

≺ 1198672

119878

4 Multicriteria Decision-Making ModelsBased on Comparisons and DistanceMeasures of HFLTSs

In this section two new methods are presented for rankingand choice from a set of alternatives in the framework ofmulticriteria decision-making using linguistic informationOne is based on the comparisons and preference relationsof HFLTSs and the other is based on the distance measureof HFLTSs We adopt Example 5 in [56] (Example 25 in ourpaper) to illustrate the detailed processes of the twomethods

Example 25 ([see [56]) Let 119883 = 1199091 1199092 1199093 be a set ofalternatives 119862 = 1198881 1198882 1198883 a set of criteria defined for eachalternative and 119878 = 1199040 n (nothing) 1199041 vl (very low)1199042 l (low) 1199043 m (medium) 1199044 h (high) 1199045

vh (very high) 1199046 p (perfect) the linguistic term set that isused to generate the linguistic expressions The assessmentsthat are provided in such a problem are shown in Table 1 andthey are transformed into HFLTSs as shown in Table 2

41 Multicriteria Decision-Making Based onthe Comparisons of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe preference degrees between all the alternatives 119909119895 (119895 =

1 2 3)Considering criterion 1198881 119867

1199091

119878= 1199041 1199042 1199043 119867

1199092

119878=

1199042 1199043 and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees

about criterion 1198881 calculated using the comparisonmethod ofHFLTSs as described in Section 32 are 1199011198881(1199091 gt 1199092) = 171199011198881(1199091 = 1199092) = 27 1199011198881(1199091 lt 1199092) = 47 1199011198881(1199091 gt 1199093) =

027 1199011198881(1199091 = 1199093) = 027 1199011198881(1199091 lt 1199093) = 2727 1199011198881(1199092 gt

1199093) = 015 1199011198881(1199092 = 1199093) = 015 1199011198881(1199092 lt 1199093) = 1515

Table 1 Assessments that are provided for the decision problem

1198881

1198882

1198883

1199091

Between vl and m Between h and vh h1199092

Between l and m m Lower than l1199093

Greater than h Between vl and l Greater than h

Table 2 Assessments transformed into HFLTSs

1198881

1198882

1198883

1199091

1199041 1199042 1199043 119904

4 1199045 119904

4

1199092

1199042 1199043 119904

3 119904

0 1199041 1199042

1199093

1199044 1199045 1199046 119904

1 1199042 119904

4 1199045 1199046

Considering criterion 1198882 1198671199091

119878= 1199044 1199045 119867

1199092

119878= 1199043

and 1198671199093

119878= 1199041 1199042 so the preference degrees about criterion

1198882 calculated using the comparison method of HFLTSs asdescribed in Section 32 are 1199011198882(1199091 gt 1199092) = 33 1199011198882(1199091 =

1199092) = 03 1199011198882(1199091 lt 1199092) = 03 1199011198882(1199091 gt 1199093) = 12121199011198882(1199091 = 1199093) = 012 1199011198882(1199091 lt 1199093) = 012 1199011198882(1199092 gt 1199093) =

33 1199011198882(1199092 = 1199093) = 03 1199011198882(1199092 lt 1199093) = 03Considering criterion 1198883 119867

1199091

119878= 1199044 119867

1199092

119878= 1199040 1199041 1199042

and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees about

criterion 1198883 calculated using the comparison method ofHFLTSs as described in Section 32 are 1199011198883(1199091 gt 1199092) = 991199011198883(1199091 = 1199092) = 09 1199011198883(1199091 lt 1199092) = 09 1199011198883(1199091 gt 1199093) = 041199011198883(1199091 = 1199093) = 14 1199011198883(1199091 lt 1199093) = 34 1199011198883(1199092 gt 1199093) =

036 1199011198883(1199092 = 1199093) = 036 1199011198883(1199092 lt 1199093) = 3636

Step 2 Aggregate the preference relations using the weightedaverage method 119901(119909119895 gt 119909119896) = sum(119908119894 times 119901119888119894(119909119895 gt 119909119896))119901(119909119895 = 119909119896) = sum(119908119894 times 119901119888i(119909119895 = 119909119896)) and 119901(119909119895 lt 119909119896) =

sum(119908119894 times 119901119888119894(119909119895 lt 119909119896)) where 119908119894 is the weight of criterion 119888119894and sum(119908119894) = 1 In this paper 119908119894 = 13 119894 = 1 2 3 Thusthe final preference relations are (1199091 gt 1199092) = 1521 119901(1199091 =

1199092) = 221 119901(1199091 lt 1199092) = 421 119901(1199091 gt 1199093) = 13119901(1199091 = 1199093) = 112 119901(1199091 lt 1199093) = 712 119901(1199092 gt 1199093) = 13119901(1199092 = 1199093) = 0 119901(1199092 lt 1199093) = 23

Step 3 Rank the alternatives using the nondominance choicedegree method as described in [56] From the results ofStep 2 it can be easily obtained that

119875119878

119863= (

minus

11

21

0

0 minus 0

1

4

1

3

minus

) (16)

Thus NDD1 = min(1 minus 0) (1 minus 14) = 34 NDD2 =

min(1 minus 1121) (1 minus 13) = 1021 and NDD3 = min(1 minus

0) (1 minus 0) = 1 Finally the ranking of alternatives is 1199093 gt

1199091 gt 1199092

6 Mathematical Problems in Engineering

Table 3 Average values of the assessments

1198881

1198882

1198883

1199091

2 45 41199092

25 3 11199093

5 15 5

Table 4 Aggregation results of each alternative

1199091

1199092

1199093

Aggregation result 35 22 38

42 Multicriteria Decision-Making Based onthe Distance Measures of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe average values of HFLTSs for all the alternatives 119909119895 (119895 =

1 2 3) The results are shown in Table 3

Step 2 Aggregate the average values using the weightedaverage method The results are shown in Table 4

Step 3 Rank the alternatives using the distance measuremethod Thus the ranking of alternatives is 1199093

03

≻ 1199091

13

≻ 1199092

43 Results Analysis In [56] the ranking of alternatives is1199091 ≻ 1199093 ≻ 1199092 while both methods in this paper are1199093 ≻ 1199091 ≻ 1199092 Note that the practical decision-makingproblem is quite different from other applications where well-establishedmeasures can be used to quantify the performancefor validation In decision-making usually there is no groundtruth data or quantitativemeasures to assess the performanceof amethod [37]This is why ldquoplausibilityrdquo is used rather thanldquovalidationrdquo Here we analyze the original assessments abouteach criterion of alternatives 1199091 and 1199093 Considering criterion1198881 the original assessments of 1199091 and 1199093 are ldquobetween vland mrdquo and ldquogreater than hrdquo respectively so it is obviously1199093 ≻ 1199091 about criterion 1198881 Considering criterion 1198882 theoriginal assessments of 1199091 and 1199093 are ldquobetween h and vhrdquoand ldquobetween vl and lrdquo respectively so this time 1199091 ≻ 1199093Considering criterion 1198883 the original assessments of 1199091 and1199093 are ldquohrdquo and ldquogreater than hrdquo respectively so 1199093 ≻ 1199091 againSummarily 1199093 ≻ 1199091 occurs twice while 1199091 ≻ 1199093 only onceThus we believe that our result is more plausible

5 Conclusion

The comparison and distance measure of HFLTSs are fun-damentally important in many decision-making problemsunder hesitant fuzzy linguistic environments From an exam-ple we found that there existed two defects when comparingHFLTSs using the previous preference degree method Byanalyzing the definition of an HFLTS a new comparisonmethod based on pairwise comparisons of each linguisticterm in the two HFLTSs has been put forward This compar-ison method does not need the assumption that the valuesin all HFLTSs are arranged in an increasing order and two

HFLTSs have the same length when comparing them Thenwe have defined a distancemeasuremethod betweenHFLTSsbased on pairwise comparisons Further we have proved thatthis distance is equal to the distance of the average values ofHFLTSs which makes the distance measure much simplerFinally two new methods for multicriteria decision-makingin which experts provide their assessments by HFLTSs havebeen proposedThe encouraging results demonstrate that ourmethods in this paper are more reasonable

In the future the application of HFLTSs to groupdecision-making problems will be explored We will alsoinvestigate how to obtain the weights of criteria underhesitant fuzzy linguistic environments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[2] M Mizumoto and K Tanaka ldquoSome properties of fuzzy sets oftype 2rdquo Information andControl vol 31 no 4 pp 312ndash340 1976

[3] J M Mendel and R I B John ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[4] J M Mendel and H Wu ldquoType-2 fuzzistics for symmetricinterval type-2 fuzzy sets part 1 forward problemsrdquo IEEETransactions on Fuzzy Systems vol 14 no 6 pp 781ndash792 2006

[5] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[6] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[7] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[8] K T Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[9] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[10] H Bustince and P Burillo ldquoVague sets are intuitionistic fuzzysetsrdquo Fuzzy Sets and Systems vol 79 no 3 pp 403ndash405 1996

[11] R R Yager ldquoOn the theory of bagsrdquo International Journal ofGeneral Systems vol 13 no 1 pp 23ndash37 1986

[12] S Miyamoto ldquoRemarks on basics of fuzzy sets and fuzzymultisetsrdquo Fuzzy Sets and Systems vol 156 no 3 pp 427ndash4312005

[13] J M Garibaldi M Jaroszewski and S Musikasuwan ldquoNonsta-tionary fuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 16no 4 pp 1072ndash1086 2008

[14] D Li J Han X Shi and M C Chan ldquoKnowledge representa-tion and discovery based on linguistic atomsrdquoKnowledge-BasedSystems vol 10 no 7 pp 431ndash440 1998

[15] D Li C Liu andWGan ldquoAnew cognitivemodel cloudmodelrdquoInternational Journal of Intelligent Systems vol 24 no 3 pp357ndash375 2009

Mathematical Problems in Engineering 7

[16] X Yang L Zeng F Luo and S Wang ldquoCloud hierarchicalanalysisrdquo Journal of Information amp Computational Science vol7 no 12 pp 2468ndash2477 2010

[17] X Yang L Zeng and R Zhang ldquoCloud Delphi methodrdquoInternational Journal of Uncertainty Fuzziness and Knowlege-Based Systems vol 20 no 1 pp 77ndash97 2012

[18] X Yang L Yan and L Zeng ldquoHow to handle uncertaintiesin AHP the Cloud Delphi hierarchical analysisrdquo InformationSciences vol 222 pp 384ndash404 2013

[19] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[20] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju Island Korea August 2009

[21] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 3 pp 199ndash249 1975

[22] L A Zadeh ldquoThe concept of a linguistic variable and its appli-cation to approximate reasoningmdashIIrdquo Information Sciences vol4 pp 301ndash357 1975

[23] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIIIrdquo Information Sciences vol9 no 1 pp 43ndash80 1975

[24] J Ma D Ruan Y Xu and G Zhang ldquoA fuzzy-set approach totreat determinacy and consistency of linguistic terms in multi-criteria decision makingrdquo International Journal of ApproximateReasoning vol 44 no 2 pp 165ndash181 2007

[25] G Bordogna and G Pasi ldquoA fuzzy linguistic approach general-izing boolean information retrieval amodel and its evaluationrdquoJournal of the American Society for Information Science vol 44no 2 pp 70ndash82 1993

[26] G Bordogna and G Pasi ldquoAn ordinal information retrievalmodelrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 9 supplement 1 2001

[27] J Kacprzyk and S Zadrozny ldquoLinguistic database summariesand their protoforms towards natural language based knowl-edge discovery toolsrdquo Information Sciences vol 173 no 4 pp281ndash304 2005

[28] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[29] H Ishibuchi T Nakashima and M Nii Classification andModeling with Linguistic Information Granules AdvancedApproaches to Linguistic Data Mining Springer Berlin Ger-many 2004

[30] R Degani and G Bortolan ldquoThe problem of linguistic approx-imation in clinical decision makingrdquo International Journal ofApproximate Reasoning vol 2 no 2 pp 143ndash162 1988

[31] E Sanchez ldquoTruth-qualification and fuzzy relations in naturallanguages application to medical diagnosisrdquo Fuzzy Sets andSystems vol 84 no 2 pp 155ndash167 1996

[32] H Lee ldquoApplying fuzzy set theory to evaluate the rate ofaggregative risk in software developmentrdquo Fuzzy Sets andSystems vol 79 no 3 pp 323ndash336 1996

[33] L Martınez ldquoSensory evaluation based on linguistic decisionanalysisrdquo International Journal of Approximate Reasoning vol44 no 2 pp 148ndash164 2007

[34] L Martınez J Liu D Ruan and J Yang ldquoDealing with het-erogeneous information in engineering evaluation processesrdquoInformation Sciences vol 177 no 7 pp 1533ndash1542 2007

[35] J Lu Y Zhu X Zeng L Koehl J Ma and G Zhang ldquoA linguis-tic multi-criteria group decision support system for fabric handevaluationrdquo FuzzyOptimization andDecisionMaking vol 8 no4 pp 395ndash413 2009

[36] R de Andres M Espinilla and L Martınez ldquoAn extendedhierarchical linguistic model for managing integral evaluationrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 486ndash500 2010

[37] D Wu and J M Mendel ldquoComputing with words for hierar-chical decision making applied to evaluating a weapon systemrdquoIEEE Transactions on Fuzzy Systems vol 18 no 3 pp 441ndash4602010

[38] F Herrera and E Herrera-Viedma ldquoChoice functions andmechanisms for linguistic preference relationsrdquo European Jour-nal of Operational Research vol 120 no 1 pp 144ndash161 2000

[39] F Herrera E Herrera-Viedma and L Martınez ldquoA fusionapproach for managingmulti-granularity linguistic term sets indecision makingrdquo Fuzzy Sets and Systems vol 114 no 1 pp 43ndash58 2000

[40] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[41] B Arfi ldquoFuzzy decisionmaking in politics a linguistic fuzzy-setapproachrdquo Political Analysis vol 13 no 1 pp 23ndash56 2005

[42] V Huynh and Y Nakamori ldquoA satisfactory-oriented approachto multiexpert decision-making with linguistic assessmentsrdquoIEEE Transactions on Systems Man and Cybernetics B Cyber-netics vol 35 no 2 pp 184ndash196 2005

[43] D Ben-Arieh and Z Chen ldquoLinguistic group decision-makingopinion aggregation and measures of consensusrdquo Fuzzy Opti-mization and Decision Making vol 5 no 4 pp 371ndash386 2006

[44] J L Garcıa-Lapresta ldquoA general class of simple majority deci-sion rules based on linguistic opinionsrdquo Information Sciencesvol 176 no 4 pp 352ndash365 2006

[45] J Wang and J Hao ldquoFuzzy linguistic PERTrdquo IEEE Transactionson Fuzzy Systems vol 15 no 2 pp 133ndash144 2007

[46] T C Wang and Y H Chen ldquoApplying fuzzy linguistic prefer-ence relations to the improvement of consistency of fuzzy AHPrdquoInformation Sciences vol 178 no 19 pp 3755ndash3765 2008

[47] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[48] P D Liu ldquoA novel method for hybridmultiple attribute decisionmakingrdquo Knowledge-Based Systems vol 22 no 5 pp 388ndash3912009

[49] Y Dong W Hong Y Xu and S Yu ldquoSelecting the individualnumerical scale and prioritization method in the analytichierarchy process a 2-Tuple fuzzy linguistic approachrdquo IEEETransactions on Fuzzy Systems vol 19 no 1 pp 13ndash25 2011

[50] L A Zadeh ldquoFuzzy logic = computing with wordsrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 103ndash111 1996

[51] J M Mendel ldquoFuzzy sets for words a new beginningrdquo inProceedings of the IEEE International conference on FuzzySystems pp 37ndash42 St Louis Mo USA May 2003

[52] J MMendel ldquoComputing with words and its relationships withfuzzisticsrdquo Information Sciences vol 177 no 4 pp 988ndash10062007

[53] F Liu and J M Mendel ldquoEncoding words into interval type-2 fuzzy sets using an interval approachrdquo IEEE Transactions onFuzzy Systems vol 16 no 6 pp 1503ndash1521 2008

8 Mathematical Problems in Engineering

[54] S Coupland J M Mendel and D Wu ldquoEnhanced intervalapproach for encoding words into interval type-2 fuzzy sets andconvergence of the word FOUsrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10) pp1ndash8 Barcelona Spain July 2010

[55] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[56] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[57] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[58] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[59] W Wang and X Xin ldquoDistance measure between intuitionisticfuzzy setsrdquo Pattern Recognition Letters vol 26 no 13 pp 2063ndash2069 2005

[60] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[61] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[62] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[63] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[64] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[65] Y Xu and H Wang ldquoDistance measure for linguistic decisionmakingrdquo Systems Engineering Procedia vol 1 pp 450ndash456 2011

[66] Y M Wang J B Yang and D L Xu ldquoA preference aggregationmethod through the estimation of utility intervalsrdquo Computersand Operations Research vol 32 no 8 pp 2027ndash2049 2005

[67] H Liao Z Xu and X Zeng ldquoDistance and similarity measuresfor hesitant fuzzy linguistic term sets and their application inmulti-criteria decision makingrdquo Information Sciences vol 271pp 125ndash142 2014

[68] B Zhu Z Xu andMXia ldquoHesitant fuzzy geometric Bonferronimeansrdquo Information Sciences vol 205 pp 72ndash85 2012

[69] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[70] X Gu Y Wang and B Yang ldquoA method for hesitant fuzzymultiple attribute decision making and its application to riskinvestmentrdquo Journal of Convergence Information Technologyvol 6 no 6 pp 282ndash287 2011

[71] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013

[72] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012

[73] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Considering Example 17 we have 119889(1198671

119878 1198672

119878) =

Aver(1198671119878) minus Aver(1198672

119878) = 2 minus 35 = minus15

By Theorem 23 we can easily obtain the following corol-lary

Corollary 24 Letting 1198671

119878 1198672119878and 119867

3

119878be three HFLTSs on S

then

(1) (|119878| minus 1) le 119889(1198671

119878 1198672

119878) le (|119878| minus 1)

(2) 119889(1198671

119878 1198672

119878) = minus119889(119867

2

119878 1198671

119878)

(3) 119889(1198671

119878 1198673

119878) = 119889(119867

1

119878 1198672

119878) + 119889(119867

2

119878 1198673

119878)

Proof They are straightforward and thus omitted

If 119889(1198671

119878 1198672

119878) gt 0 (or Aver(1198671

119878) gt Aver(1198672

119878)) then we

say that 1198671119878is superior to 119867

2

119878with the distance of 119889(1198671

119878 1198672

119878)

denoted by 1198671

119878

119889(1198671

1198781198672

119878)

≻ 1198672

119878 if 119889(1198671

119878 1198672

119878) = 0 (or Aver(1198671

119878) =

Aver(1198672119878)) then we say that 1198671

119878is indifferent to 119867

2

119878 denoted

by 1198671

119878sim 1198672

119878 if 119889(1198671

119878 1198672

119878) lt 0 (or Aver(1198671

119878) lt Aver(1198672

119878))

then we say that 1198671

119878is inferior to 119867

2

119878with the distance of

119889(1198671

119878 1198672

119878) denoted by 119867

1

119878

119889(1198671

1198781198672

119878)

≺ 1198672

119878

4 Multicriteria Decision-Making ModelsBased on Comparisons and DistanceMeasures of HFLTSs

In this section two new methods are presented for rankingand choice from a set of alternatives in the framework ofmulticriteria decision-making using linguistic informationOne is based on the comparisons and preference relationsof HFLTSs and the other is based on the distance measureof HFLTSs We adopt Example 5 in [56] (Example 25 in ourpaper) to illustrate the detailed processes of the twomethods

Example 25 ([see [56]) Let 119883 = 1199091 1199092 1199093 be a set ofalternatives 119862 = 1198881 1198882 1198883 a set of criteria defined for eachalternative and 119878 = 1199040 n (nothing) 1199041 vl (very low)1199042 l (low) 1199043 m (medium) 1199044 h (high) 1199045

vh (very high) 1199046 p (perfect) the linguistic term set that isused to generate the linguistic expressions The assessmentsthat are provided in such a problem are shown in Table 1 andthey are transformed into HFLTSs as shown in Table 2

41 Multicriteria Decision-Making Based onthe Comparisons of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe preference degrees between all the alternatives 119909119895 (119895 =

1 2 3)Considering criterion 1198881 119867

1199091

119878= 1199041 1199042 1199043 119867

1199092

119878=

1199042 1199043 and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees

about criterion 1198881 calculated using the comparisonmethod ofHFLTSs as described in Section 32 are 1199011198881(1199091 gt 1199092) = 171199011198881(1199091 = 1199092) = 27 1199011198881(1199091 lt 1199092) = 47 1199011198881(1199091 gt 1199093) =

027 1199011198881(1199091 = 1199093) = 027 1199011198881(1199091 lt 1199093) = 2727 1199011198881(1199092 gt

1199093) = 015 1199011198881(1199092 = 1199093) = 015 1199011198881(1199092 lt 1199093) = 1515

Table 1 Assessments that are provided for the decision problem

1198881

1198882

1198883

1199091

Between vl and m Between h and vh h1199092

Between l and m m Lower than l1199093

Greater than h Between vl and l Greater than h

Table 2 Assessments transformed into HFLTSs

1198881

1198882

1198883

1199091

1199041 1199042 1199043 119904

4 1199045 119904

4

1199092

1199042 1199043 119904

3 119904

0 1199041 1199042

1199093

1199044 1199045 1199046 119904

1 1199042 119904

4 1199045 1199046

Considering criterion 1198882 1198671199091

119878= 1199044 1199045 119867

1199092

119878= 1199043

and 1198671199093

119878= 1199041 1199042 so the preference degrees about criterion

1198882 calculated using the comparison method of HFLTSs asdescribed in Section 32 are 1199011198882(1199091 gt 1199092) = 33 1199011198882(1199091 =

1199092) = 03 1199011198882(1199091 lt 1199092) = 03 1199011198882(1199091 gt 1199093) = 12121199011198882(1199091 = 1199093) = 012 1199011198882(1199091 lt 1199093) = 012 1199011198882(1199092 gt 1199093) =

33 1199011198882(1199092 = 1199093) = 03 1199011198882(1199092 lt 1199093) = 03Considering criterion 1198883 119867

1199091

119878= 1199044 119867

1199092

119878= 1199040 1199041 1199042

and 1198671199093

119878= 1199044 1199045 1199046 so the preference degrees about

criterion 1198883 calculated using the comparison method ofHFLTSs as described in Section 32 are 1199011198883(1199091 gt 1199092) = 991199011198883(1199091 = 1199092) = 09 1199011198883(1199091 lt 1199092) = 09 1199011198883(1199091 gt 1199093) = 041199011198883(1199091 = 1199093) = 14 1199011198883(1199091 lt 1199093) = 34 1199011198883(1199092 gt 1199093) =

036 1199011198883(1199092 = 1199093) = 036 1199011198883(1199092 lt 1199093) = 3636

Step 2 Aggregate the preference relations using the weightedaverage method 119901(119909119895 gt 119909119896) = sum(119908119894 times 119901119888119894(119909119895 gt 119909119896))119901(119909119895 = 119909119896) = sum(119908119894 times 119901119888i(119909119895 = 119909119896)) and 119901(119909119895 lt 119909119896) =

sum(119908119894 times 119901119888119894(119909119895 lt 119909119896)) where 119908119894 is the weight of criterion 119888119894and sum(119908119894) = 1 In this paper 119908119894 = 13 119894 = 1 2 3 Thusthe final preference relations are (1199091 gt 1199092) = 1521 119901(1199091 =

1199092) = 221 119901(1199091 lt 1199092) = 421 119901(1199091 gt 1199093) = 13119901(1199091 = 1199093) = 112 119901(1199091 lt 1199093) = 712 119901(1199092 gt 1199093) = 13119901(1199092 = 1199093) = 0 119901(1199092 lt 1199093) = 23

Step 3 Rank the alternatives using the nondominance choicedegree method as described in [56] From the results ofStep 2 it can be easily obtained that

119875119878

119863= (

minus

11

21

0

0 minus 0

1

4

1

3

minus

) (16)

Thus NDD1 = min(1 minus 0) (1 minus 14) = 34 NDD2 =

min(1 minus 1121) (1 minus 13) = 1021 and NDD3 = min(1 minus

0) (1 minus 0) = 1 Finally the ranking of alternatives is 1199093 gt

1199091 gt 1199092

6 Mathematical Problems in Engineering

Table 3 Average values of the assessments

1198881

1198882

1198883

1199091

2 45 41199092

25 3 11199093

5 15 5

Table 4 Aggregation results of each alternative

1199091

1199092

1199093

Aggregation result 35 22 38

42 Multicriteria Decision-Making Based onthe Distance Measures of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe average values of HFLTSs for all the alternatives 119909119895 (119895 =

1 2 3) The results are shown in Table 3

Step 2 Aggregate the average values using the weightedaverage method The results are shown in Table 4

Step 3 Rank the alternatives using the distance measuremethod Thus the ranking of alternatives is 1199093

03

≻ 1199091

13

≻ 1199092

43 Results Analysis In [56] the ranking of alternatives is1199091 ≻ 1199093 ≻ 1199092 while both methods in this paper are1199093 ≻ 1199091 ≻ 1199092 Note that the practical decision-makingproblem is quite different from other applications where well-establishedmeasures can be used to quantify the performancefor validation In decision-making usually there is no groundtruth data or quantitativemeasures to assess the performanceof amethod [37]This is why ldquoplausibilityrdquo is used rather thanldquovalidationrdquo Here we analyze the original assessments abouteach criterion of alternatives 1199091 and 1199093 Considering criterion1198881 the original assessments of 1199091 and 1199093 are ldquobetween vland mrdquo and ldquogreater than hrdquo respectively so it is obviously1199093 ≻ 1199091 about criterion 1198881 Considering criterion 1198882 theoriginal assessments of 1199091 and 1199093 are ldquobetween h and vhrdquoand ldquobetween vl and lrdquo respectively so this time 1199091 ≻ 1199093Considering criterion 1198883 the original assessments of 1199091 and1199093 are ldquohrdquo and ldquogreater than hrdquo respectively so 1199093 ≻ 1199091 againSummarily 1199093 ≻ 1199091 occurs twice while 1199091 ≻ 1199093 only onceThus we believe that our result is more plausible

5 Conclusion

The comparison and distance measure of HFLTSs are fun-damentally important in many decision-making problemsunder hesitant fuzzy linguistic environments From an exam-ple we found that there existed two defects when comparingHFLTSs using the previous preference degree method Byanalyzing the definition of an HFLTS a new comparisonmethod based on pairwise comparisons of each linguisticterm in the two HFLTSs has been put forward This compar-ison method does not need the assumption that the valuesin all HFLTSs are arranged in an increasing order and two

HFLTSs have the same length when comparing them Thenwe have defined a distancemeasuremethod betweenHFLTSsbased on pairwise comparisons Further we have proved thatthis distance is equal to the distance of the average values ofHFLTSs which makes the distance measure much simplerFinally two new methods for multicriteria decision-makingin which experts provide their assessments by HFLTSs havebeen proposedThe encouraging results demonstrate that ourmethods in this paper are more reasonable

In the future the application of HFLTSs to groupdecision-making problems will be explored We will alsoinvestigate how to obtain the weights of criteria underhesitant fuzzy linguistic environments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[2] M Mizumoto and K Tanaka ldquoSome properties of fuzzy sets oftype 2rdquo Information andControl vol 31 no 4 pp 312ndash340 1976

[3] J M Mendel and R I B John ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[4] J M Mendel and H Wu ldquoType-2 fuzzistics for symmetricinterval type-2 fuzzy sets part 1 forward problemsrdquo IEEETransactions on Fuzzy Systems vol 14 no 6 pp 781ndash792 2006

[5] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[6] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[7] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[8] K T Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[9] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[10] H Bustince and P Burillo ldquoVague sets are intuitionistic fuzzysetsrdquo Fuzzy Sets and Systems vol 79 no 3 pp 403ndash405 1996

[11] R R Yager ldquoOn the theory of bagsrdquo International Journal ofGeneral Systems vol 13 no 1 pp 23ndash37 1986

[12] S Miyamoto ldquoRemarks on basics of fuzzy sets and fuzzymultisetsrdquo Fuzzy Sets and Systems vol 156 no 3 pp 427ndash4312005

[13] J M Garibaldi M Jaroszewski and S Musikasuwan ldquoNonsta-tionary fuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 16no 4 pp 1072ndash1086 2008

[14] D Li J Han X Shi and M C Chan ldquoKnowledge representa-tion and discovery based on linguistic atomsrdquoKnowledge-BasedSystems vol 10 no 7 pp 431ndash440 1998

[15] D Li C Liu andWGan ldquoAnew cognitivemodel cloudmodelrdquoInternational Journal of Intelligent Systems vol 24 no 3 pp357ndash375 2009

Mathematical Problems in Engineering 7

[16] X Yang L Zeng F Luo and S Wang ldquoCloud hierarchicalanalysisrdquo Journal of Information amp Computational Science vol7 no 12 pp 2468ndash2477 2010

[17] X Yang L Zeng and R Zhang ldquoCloud Delphi methodrdquoInternational Journal of Uncertainty Fuzziness and Knowlege-Based Systems vol 20 no 1 pp 77ndash97 2012

[18] X Yang L Yan and L Zeng ldquoHow to handle uncertaintiesin AHP the Cloud Delphi hierarchical analysisrdquo InformationSciences vol 222 pp 384ndash404 2013

[19] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[20] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju Island Korea August 2009

[21] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 3 pp 199ndash249 1975

[22] L A Zadeh ldquoThe concept of a linguistic variable and its appli-cation to approximate reasoningmdashIIrdquo Information Sciences vol4 pp 301ndash357 1975

[23] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIIIrdquo Information Sciences vol9 no 1 pp 43ndash80 1975

[24] J Ma D Ruan Y Xu and G Zhang ldquoA fuzzy-set approach totreat determinacy and consistency of linguistic terms in multi-criteria decision makingrdquo International Journal of ApproximateReasoning vol 44 no 2 pp 165ndash181 2007

[25] G Bordogna and G Pasi ldquoA fuzzy linguistic approach general-izing boolean information retrieval amodel and its evaluationrdquoJournal of the American Society for Information Science vol 44no 2 pp 70ndash82 1993

[26] G Bordogna and G Pasi ldquoAn ordinal information retrievalmodelrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 9 supplement 1 2001

[27] J Kacprzyk and S Zadrozny ldquoLinguistic database summariesand their protoforms towards natural language based knowl-edge discovery toolsrdquo Information Sciences vol 173 no 4 pp281ndash304 2005

[28] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[29] H Ishibuchi T Nakashima and M Nii Classification andModeling with Linguistic Information Granules AdvancedApproaches to Linguistic Data Mining Springer Berlin Ger-many 2004

[30] R Degani and G Bortolan ldquoThe problem of linguistic approx-imation in clinical decision makingrdquo International Journal ofApproximate Reasoning vol 2 no 2 pp 143ndash162 1988

[31] E Sanchez ldquoTruth-qualification and fuzzy relations in naturallanguages application to medical diagnosisrdquo Fuzzy Sets andSystems vol 84 no 2 pp 155ndash167 1996

[32] H Lee ldquoApplying fuzzy set theory to evaluate the rate ofaggregative risk in software developmentrdquo Fuzzy Sets andSystems vol 79 no 3 pp 323ndash336 1996

[33] L Martınez ldquoSensory evaluation based on linguistic decisionanalysisrdquo International Journal of Approximate Reasoning vol44 no 2 pp 148ndash164 2007

[34] L Martınez J Liu D Ruan and J Yang ldquoDealing with het-erogeneous information in engineering evaluation processesrdquoInformation Sciences vol 177 no 7 pp 1533ndash1542 2007

[35] J Lu Y Zhu X Zeng L Koehl J Ma and G Zhang ldquoA linguis-tic multi-criteria group decision support system for fabric handevaluationrdquo FuzzyOptimization andDecisionMaking vol 8 no4 pp 395ndash413 2009

[36] R de Andres M Espinilla and L Martınez ldquoAn extendedhierarchical linguistic model for managing integral evaluationrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 486ndash500 2010

[37] D Wu and J M Mendel ldquoComputing with words for hierar-chical decision making applied to evaluating a weapon systemrdquoIEEE Transactions on Fuzzy Systems vol 18 no 3 pp 441ndash4602010

[38] F Herrera and E Herrera-Viedma ldquoChoice functions andmechanisms for linguistic preference relationsrdquo European Jour-nal of Operational Research vol 120 no 1 pp 144ndash161 2000

[39] F Herrera E Herrera-Viedma and L Martınez ldquoA fusionapproach for managingmulti-granularity linguistic term sets indecision makingrdquo Fuzzy Sets and Systems vol 114 no 1 pp 43ndash58 2000

[40] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[41] B Arfi ldquoFuzzy decisionmaking in politics a linguistic fuzzy-setapproachrdquo Political Analysis vol 13 no 1 pp 23ndash56 2005

[42] V Huynh and Y Nakamori ldquoA satisfactory-oriented approachto multiexpert decision-making with linguistic assessmentsrdquoIEEE Transactions on Systems Man and Cybernetics B Cyber-netics vol 35 no 2 pp 184ndash196 2005

[43] D Ben-Arieh and Z Chen ldquoLinguistic group decision-makingopinion aggregation and measures of consensusrdquo Fuzzy Opti-mization and Decision Making vol 5 no 4 pp 371ndash386 2006

[44] J L Garcıa-Lapresta ldquoA general class of simple majority deci-sion rules based on linguistic opinionsrdquo Information Sciencesvol 176 no 4 pp 352ndash365 2006

[45] J Wang and J Hao ldquoFuzzy linguistic PERTrdquo IEEE Transactionson Fuzzy Systems vol 15 no 2 pp 133ndash144 2007

[46] T C Wang and Y H Chen ldquoApplying fuzzy linguistic prefer-ence relations to the improvement of consistency of fuzzy AHPrdquoInformation Sciences vol 178 no 19 pp 3755ndash3765 2008

[47] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[48] P D Liu ldquoA novel method for hybridmultiple attribute decisionmakingrdquo Knowledge-Based Systems vol 22 no 5 pp 388ndash3912009

[49] Y Dong W Hong Y Xu and S Yu ldquoSelecting the individualnumerical scale and prioritization method in the analytichierarchy process a 2-Tuple fuzzy linguistic approachrdquo IEEETransactions on Fuzzy Systems vol 19 no 1 pp 13ndash25 2011

[50] L A Zadeh ldquoFuzzy logic = computing with wordsrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 103ndash111 1996

[51] J M Mendel ldquoFuzzy sets for words a new beginningrdquo inProceedings of the IEEE International conference on FuzzySystems pp 37ndash42 St Louis Mo USA May 2003

[52] J MMendel ldquoComputing with words and its relationships withfuzzisticsrdquo Information Sciences vol 177 no 4 pp 988ndash10062007

[53] F Liu and J M Mendel ldquoEncoding words into interval type-2 fuzzy sets using an interval approachrdquo IEEE Transactions onFuzzy Systems vol 16 no 6 pp 1503ndash1521 2008

8 Mathematical Problems in Engineering

[54] S Coupland J M Mendel and D Wu ldquoEnhanced intervalapproach for encoding words into interval type-2 fuzzy sets andconvergence of the word FOUsrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10) pp1ndash8 Barcelona Spain July 2010

[55] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[56] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[57] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[58] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[59] W Wang and X Xin ldquoDistance measure between intuitionisticfuzzy setsrdquo Pattern Recognition Letters vol 26 no 13 pp 2063ndash2069 2005

[60] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[61] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[62] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[63] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[64] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[65] Y Xu and H Wang ldquoDistance measure for linguistic decisionmakingrdquo Systems Engineering Procedia vol 1 pp 450ndash456 2011

[66] Y M Wang J B Yang and D L Xu ldquoA preference aggregationmethod through the estimation of utility intervalsrdquo Computersand Operations Research vol 32 no 8 pp 2027ndash2049 2005

[67] H Liao Z Xu and X Zeng ldquoDistance and similarity measuresfor hesitant fuzzy linguistic term sets and their application inmulti-criteria decision makingrdquo Information Sciences vol 271pp 125ndash142 2014

[68] B Zhu Z Xu andMXia ldquoHesitant fuzzy geometric Bonferronimeansrdquo Information Sciences vol 205 pp 72ndash85 2012

[69] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[70] X Gu Y Wang and B Yang ldquoA method for hesitant fuzzymultiple attribute decision making and its application to riskinvestmentrdquo Journal of Convergence Information Technologyvol 6 no 6 pp 282ndash287 2011

[71] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013

[72] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012

[73] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 3 Average values of the assessments

1198881

1198882

1198883

1199091

2 45 41199092

25 3 11199093

5 15 5

Table 4 Aggregation results of each alternative

1199091

1199092

1199093

Aggregation result 35 22 38

42 Multicriteria Decision-Making Based onthe Distance Measures of HFLTSs

Step 1 Considering each criterion 119888119894 (119894 = 1 2 3) calculatethe average values of HFLTSs for all the alternatives 119909119895 (119895 =

1 2 3) The results are shown in Table 3

Step 2 Aggregate the average values using the weightedaverage method The results are shown in Table 4

Step 3 Rank the alternatives using the distance measuremethod Thus the ranking of alternatives is 1199093

03

≻ 1199091

13

≻ 1199092

43 Results Analysis In [56] the ranking of alternatives is1199091 ≻ 1199093 ≻ 1199092 while both methods in this paper are1199093 ≻ 1199091 ≻ 1199092 Note that the practical decision-makingproblem is quite different from other applications where well-establishedmeasures can be used to quantify the performancefor validation In decision-making usually there is no groundtruth data or quantitativemeasures to assess the performanceof amethod [37]This is why ldquoplausibilityrdquo is used rather thanldquovalidationrdquo Here we analyze the original assessments abouteach criterion of alternatives 1199091 and 1199093 Considering criterion1198881 the original assessments of 1199091 and 1199093 are ldquobetween vland mrdquo and ldquogreater than hrdquo respectively so it is obviously1199093 ≻ 1199091 about criterion 1198881 Considering criterion 1198882 theoriginal assessments of 1199091 and 1199093 are ldquobetween h and vhrdquoand ldquobetween vl and lrdquo respectively so this time 1199091 ≻ 1199093Considering criterion 1198883 the original assessments of 1199091 and1199093 are ldquohrdquo and ldquogreater than hrdquo respectively so 1199093 ≻ 1199091 againSummarily 1199093 ≻ 1199091 occurs twice while 1199091 ≻ 1199093 only onceThus we believe that our result is more plausible

5 Conclusion

The comparison and distance measure of HFLTSs are fun-damentally important in many decision-making problemsunder hesitant fuzzy linguistic environments From an exam-ple we found that there existed two defects when comparingHFLTSs using the previous preference degree method Byanalyzing the definition of an HFLTS a new comparisonmethod based on pairwise comparisons of each linguisticterm in the two HFLTSs has been put forward This compar-ison method does not need the assumption that the valuesin all HFLTSs are arranged in an increasing order and two

HFLTSs have the same length when comparing them Thenwe have defined a distancemeasuremethod betweenHFLTSsbased on pairwise comparisons Further we have proved thatthis distance is equal to the distance of the average values ofHFLTSs which makes the distance measure much simplerFinally two new methods for multicriteria decision-makingin which experts provide their assessments by HFLTSs havebeen proposedThe encouraging results demonstrate that ourmethods in this paper are more reasonable

In the future the application of HFLTSs to groupdecision-making problems will be explored We will alsoinvestigate how to obtain the weights of criteria underhesitant fuzzy linguistic environments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[2] M Mizumoto and K Tanaka ldquoSome properties of fuzzy sets oftype 2rdquo Information andControl vol 31 no 4 pp 312ndash340 1976

[3] J M Mendel and R I B John ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[4] J M Mendel and H Wu ldquoType-2 fuzzistics for symmetricinterval type-2 fuzzy sets part 1 forward problemsrdquo IEEETransactions on Fuzzy Systems vol 14 no 6 pp 781ndash792 2006

[5] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[6] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[7] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[8] K T Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[9] W Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[10] H Bustince and P Burillo ldquoVague sets are intuitionistic fuzzysetsrdquo Fuzzy Sets and Systems vol 79 no 3 pp 403ndash405 1996

[11] R R Yager ldquoOn the theory of bagsrdquo International Journal ofGeneral Systems vol 13 no 1 pp 23ndash37 1986

[12] S Miyamoto ldquoRemarks on basics of fuzzy sets and fuzzymultisetsrdquo Fuzzy Sets and Systems vol 156 no 3 pp 427ndash4312005

[13] J M Garibaldi M Jaroszewski and S Musikasuwan ldquoNonsta-tionary fuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 16no 4 pp 1072ndash1086 2008

[14] D Li J Han X Shi and M C Chan ldquoKnowledge representa-tion and discovery based on linguistic atomsrdquoKnowledge-BasedSystems vol 10 no 7 pp 431ndash440 1998

[15] D Li C Liu andWGan ldquoAnew cognitivemodel cloudmodelrdquoInternational Journal of Intelligent Systems vol 24 no 3 pp357ndash375 2009

Mathematical Problems in Engineering 7

[16] X Yang L Zeng F Luo and S Wang ldquoCloud hierarchicalanalysisrdquo Journal of Information amp Computational Science vol7 no 12 pp 2468ndash2477 2010

[17] X Yang L Zeng and R Zhang ldquoCloud Delphi methodrdquoInternational Journal of Uncertainty Fuzziness and Knowlege-Based Systems vol 20 no 1 pp 77ndash97 2012

[18] X Yang L Yan and L Zeng ldquoHow to handle uncertaintiesin AHP the Cloud Delphi hierarchical analysisrdquo InformationSciences vol 222 pp 384ndash404 2013

[19] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[20] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju Island Korea August 2009

[21] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 3 pp 199ndash249 1975

[22] L A Zadeh ldquoThe concept of a linguistic variable and its appli-cation to approximate reasoningmdashIIrdquo Information Sciences vol4 pp 301ndash357 1975

[23] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIIIrdquo Information Sciences vol9 no 1 pp 43ndash80 1975

[24] J Ma D Ruan Y Xu and G Zhang ldquoA fuzzy-set approach totreat determinacy and consistency of linguistic terms in multi-criteria decision makingrdquo International Journal of ApproximateReasoning vol 44 no 2 pp 165ndash181 2007

[25] G Bordogna and G Pasi ldquoA fuzzy linguistic approach general-izing boolean information retrieval amodel and its evaluationrdquoJournal of the American Society for Information Science vol 44no 2 pp 70ndash82 1993

[26] G Bordogna and G Pasi ldquoAn ordinal information retrievalmodelrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 9 supplement 1 2001

[27] J Kacprzyk and S Zadrozny ldquoLinguistic database summariesand their protoforms towards natural language based knowl-edge discovery toolsrdquo Information Sciences vol 173 no 4 pp281ndash304 2005

[28] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[29] H Ishibuchi T Nakashima and M Nii Classification andModeling with Linguistic Information Granules AdvancedApproaches to Linguistic Data Mining Springer Berlin Ger-many 2004

[30] R Degani and G Bortolan ldquoThe problem of linguistic approx-imation in clinical decision makingrdquo International Journal ofApproximate Reasoning vol 2 no 2 pp 143ndash162 1988

[31] E Sanchez ldquoTruth-qualification and fuzzy relations in naturallanguages application to medical diagnosisrdquo Fuzzy Sets andSystems vol 84 no 2 pp 155ndash167 1996

[32] H Lee ldquoApplying fuzzy set theory to evaluate the rate ofaggregative risk in software developmentrdquo Fuzzy Sets andSystems vol 79 no 3 pp 323ndash336 1996

[33] L Martınez ldquoSensory evaluation based on linguistic decisionanalysisrdquo International Journal of Approximate Reasoning vol44 no 2 pp 148ndash164 2007

[34] L Martınez J Liu D Ruan and J Yang ldquoDealing with het-erogeneous information in engineering evaluation processesrdquoInformation Sciences vol 177 no 7 pp 1533ndash1542 2007

[35] J Lu Y Zhu X Zeng L Koehl J Ma and G Zhang ldquoA linguis-tic multi-criteria group decision support system for fabric handevaluationrdquo FuzzyOptimization andDecisionMaking vol 8 no4 pp 395ndash413 2009

[36] R de Andres M Espinilla and L Martınez ldquoAn extendedhierarchical linguistic model for managing integral evaluationrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 486ndash500 2010

[37] D Wu and J M Mendel ldquoComputing with words for hierar-chical decision making applied to evaluating a weapon systemrdquoIEEE Transactions on Fuzzy Systems vol 18 no 3 pp 441ndash4602010

[38] F Herrera and E Herrera-Viedma ldquoChoice functions andmechanisms for linguistic preference relationsrdquo European Jour-nal of Operational Research vol 120 no 1 pp 144ndash161 2000

[39] F Herrera E Herrera-Viedma and L Martınez ldquoA fusionapproach for managingmulti-granularity linguistic term sets indecision makingrdquo Fuzzy Sets and Systems vol 114 no 1 pp 43ndash58 2000

[40] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[41] B Arfi ldquoFuzzy decisionmaking in politics a linguistic fuzzy-setapproachrdquo Political Analysis vol 13 no 1 pp 23ndash56 2005

[42] V Huynh and Y Nakamori ldquoA satisfactory-oriented approachto multiexpert decision-making with linguistic assessmentsrdquoIEEE Transactions on Systems Man and Cybernetics B Cyber-netics vol 35 no 2 pp 184ndash196 2005

[43] D Ben-Arieh and Z Chen ldquoLinguistic group decision-makingopinion aggregation and measures of consensusrdquo Fuzzy Opti-mization and Decision Making vol 5 no 4 pp 371ndash386 2006

[44] J L Garcıa-Lapresta ldquoA general class of simple majority deci-sion rules based on linguistic opinionsrdquo Information Sciencesvol 176 no 4 pp 352ndash365 2006

[45] J Wang and J Hao ldquoFuzzy linguistic PERTrdquo IEEE Transactionson Fuzzy Systems vol 15 no 2 pp 133ndash144 2007

[46] T C Wang and Y H Chen ldquoApplying fuzzy linguistic prefer-ence relations to the improvement of consistency of fuzzy AHPrdquoInformation Sciences vol 178 no 19 pp 3755ndash3765 2008

[47] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[48] P D Liu ldquoA novel method for hybridmultiple attribute decisionmakingrdquo Knowledge-Based Systems vol 22 no 5 pp 388ndash3912009

[49] Y Dong W Hong Y Xu and S Yu ldquoSelecting the individualnumerical scale and prioritization method in the analytichierarchy process a 2-Tuple fuzzy linguistic approachrdquo IEEETransactions on Fuzzy Systems vol 19 no 1 pp 13ndash25 2011

[50] L A Zadeh ldquoFuzzy logic = computing with wordsrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 103ndash111 1996

[51] J M Mendel ldquoFuzzy sets for words a new beginningrdquo inProceedings of the IEEE International conference on FuzzySystems pp 37ndash42 St Louis Mo USA May 2003

[52] J MMendel ldquoComputing with words and its relationships withfuzzisticsrdquo Information Sciences vol 177 no 4 pp 988ndash10062007

[53] F Liu and J M Mendel ldquoEncoding words into interval type-2 fuzzy sets using an interval approachrdquo IEEE Transactions onFuzzy Systems vol 16 no 6 pp 1503ndash1521 2008

8 Mathematical Problems in Engineering

[54] S Coupland J M Mendel and D Wu ldquoEnhanced intervalapproach for encoding words into interval type-2 fuzzy sets andconvergence of the word FOUsrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10) pp1ndash8 Barcelona Spain July 2010

[55] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[56] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[57] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[58] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[59] W Wang and X Xin ldquoDistance measure between intuitionisticfuzzy setsrdquo Pattern Recognition Letters vol 26 no 13 pp 2063ndash2069 2005

[60] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[61] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[62] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[63] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[64] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[65] Y Xu and H Wang ldquoDistance measure for linguistic decisionmakingrdquo Systems Engineering Procedia vol 1 pp 450ndash456 2011

[66] Y M Wang J B Yang and D L Xu ldquoA preference aggregationmethod through the estimation of utility intervalsrdquo Computersand Operations Research vol 32 no 8 pp 2027ndash2049 2005

[67] H Liao Z Xu and X Zeng ldquoDistance and similarity measuresfor hesitant fuzzy linguistic term sets and their application inmulti-criteria decision makingrdquo Information Sciences vol 271pp 125ndash142 2014

[68] B Zhu Z Xu andMXia ldquoHesitant fuzzy geometric Bonferronimeansrdquo Information Sciences vol 205 pp 72ndash85 2012

[69] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[70] X Gu Y Wang and B Yang ldquoA method for hesitant fuzzymultiple attribute decision making and its application to riskinvestmentrdquo Journal of Convergence Information Technologyvol 6 no 6 pp 282ndash287 2011

[71] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013

[72] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012

[73] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

[16] X Yang L Zeng F Luo and S Wang ldquoCloud hierarchicalanalysisrdquo Journal of Information amp Computational Science vol7 no 12 pp 2468ndash2477 2010

[17] X Yang L Zeng and R Zhang ldquoCloud Delphi methodrdquoInternational Journal of Uncertainty Fuzziness and Knowlege-Based Systems vol 20 no 1 pp 77ndash97 2012

[18] X Yang L Yan and L Zeng ldquoHow to handle uncertaintiesin AHP the Cloud Delphi hierarchical analysisrdquo InformationSciences vol 222 pp 384ndash404 2013

[19] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[20] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju Island Korea August 2009

[21] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 3 pp 199ndash249 1975

[22] L A Zadeh ldquoThe concept of a linguistic variable and its appli-cation to approximate reasoningmdashIIrdquo Information Sciences vol4 pp 301ndash357 1975

[23] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIIIrdquo Information Sciences vol9 no 1 pp 43ndash80 1975

[24] J Ma D Ruan Y Xu and G Zhang ldquoA fuzzy-set approach totreat determinacy and consistency of linguistic terms in multi-criteria decision makingrdquo International Journal of ApproximateReasoning vol 44 no 2 pp 165ndash181 2007

[25] G Bordogna and G Pasi ldquoA fuzzy linguistic approach general-izing boolean information retrieval amodel and its evaluationrdquoJournal of the American Society for Information Science vol 44no 2 pp 70ndash82 1993

[26] G Bordogna and G Pasi ldquoAn ordinal information retrievalmodelrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 9 supplement 1 2001

[27] J Kacprzyk and S Zadrozny ldquoLinguistic database summariesand their protoforms towards natural language based knowl-edge discovery toolsrdquo Information Sciences vol 173 no 4 pp281ndash304 2005

[28] E Herrera-Viedma and A G Lopez-Herrera ldquoA model of aninformation retrieval system with unbalanced fuzzy linguisticinformationrdquo International Journal of Intelligent Systems vol 22no 11 pp 1197ndash1214 2007

[29] H Ishibuchi T Nakashima and M Nii Classification andModeling with Linguistic Information Granules AdvancedApproaches to Linguistic Data Mining Springer Berlin Ger-many 2004

[30] R Degani and G Bortolan ldquoThe problem of linguistic approx-imation in clinical decision makingrdquo International Journal ofApproximate Reasoning vol 2 no 2 pp 143ndash162 1988

[31] E Sanchez ldquoTruth-qualification and fuzzy relations in naturallanguages application to medical diagnosisrdquo Fuzzy Sets andSystems vol 84 no 2 pp 155ndash167 1996

[32] H Lee ldquoApplying fuzzy set theory to evaluate the rate ofaggregative risk in software developmentrdquo Fuzzy Sets andSystems vol 79 no 3 pp 323ndash336 1996

[33] L Martınez ldquoSensory evaluation based on linguistic decisionanalysisrdquo International Journal of Approximate Reasoning vol44 no 2 pp 148ndash164 2007

[34] L Martınez J Liu D Ruan and J Yang ldquoDealing with het-erogeneous information in engineering evaluation processesrdquoInformation Sciences vol 177 no 7 pp 1533ndash1542 2007

[35] J Lu Y Zhu X Zeng L Koehl J Ma and G Zhang ldquoA linguis-tic multi-criteria group decision support system for fabric handevaluationrdquo FuzzyOptimization andDecisionMaking vol 8 no4 pp 395ndash413 2009

[36] R de Andres M Espinilla and L Martınez ldquoAn extendedhierarchical linguistic model for managing integral evaluationrdquoInternational Journal of Computational Intelligence Systems vol3 no 4 pp 486ndash500 2010

[37] D Wu and J M Mendel ldquoComputing with words for hierar-chical decision making applied to evaluating a weapon systemrdquoIEEE Transactions on Fuzzy Systems vol 18 no 3 pp 441ndash4602010

[38] F Herrera and E Herrera-Viedma ldquoChoice functions andmechanisms for linguistic preference relationsrdquo European Jour-nal of Operational Research vol 120 no 1 pp 144ndash161 2000

[39] F Herrera E Herrera-Viedma and L Martınez ldquoA fusionapproach for managingmulti-granularity linguistic term sets indecision makingrdquo Fuzzy Sets and Systems vol 114 no 1 pp 43ndash58 2000

[40] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[41] B Arfi ldquoFuzzy decisionmaking in politics a linguistic fuzzy-setapproachrdquo Political Analysis vol 13 no 1 pp 23ndash56 2005

[42] V Huynh and Y Nakamori ldquoA satisfactory-oriented approachto multiexpert decision-making with linguistic assessmentsrdquoIEEE Transactions on Systems Man and Cybernetics B Cyber-netics vol 35 no 2 pp 184ndash196 2005

[43] D Ben-Arieh and Z Chen ldquoLinguistic group decision-makingopinion aggregation and measures of consensusrdquo Fuzzy Opti-mization and Decision Making vol 5 no 4 pp 371ndash386 2006

[44] J L Garcıa-Lapresta ldquoA general class of simple majority deci-sion rules based on linguistic opinionsrdquo Information Sciencesvol 176 no 4 pp 352ndash365 2006

[45] J Wang and J Hao ldquoFuzzy linguistic PERTrdquo IEEE Transactionson Fuzzy Systems vol 15 no 2 pp 133ndash144 2007

[46] T C Wang and Y H Chen ldquoApplying fuzzy linguistic prefer-ence relations to the improvement of consistency of fuzzy AHPrdquoInformation Sciences vol 178 no 19 pp 3755ndash3765 2008

[47] Y Dong Y Xu and S Yu ldquoLinguistic multiperson decisionmaking based on the use ofmultiple preference relationsrdquo FuzzySets and Systems vol 160 no 5 pp 603ndash623 2009

[48] P D Liu ldquoA novel method for hybridmultiple attribute decisionmakingrdquo Knowledge-Based Systems vol 22 no 5 pp 388ndash3912009

[49] Y Dong W Hong Y Xu and S Yu ldquoSelecting the individualnumerical scale and prioritization method in the analytichierarchy process a 2-Tuple fuzzy linguistic approachrdquo IEEETransactions on Fuzzy Systems vol 19 no 1 pp 13ndash25 2011

[50] L A Zadeh ldquoFuzzy logic = computing with wordsrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 103ndash111 1996

[51] J M Mendel ldquoFuzzy sets for words a new beginningrdquo inProceedings of the IEEE International conference on FuzzySystems pp 37ndash42 St Louis Mo USA May 2003

[52] J MMendel ldquoComputing with words and its relationships withfuzzisticsrdquo Information Sciences vol 177 no 4 pp 988ndash10062007

[53] F Liu and J M Mendel ldquoEncoding words into interval type-2 fuzzy sets using an interval approachrdquo IEEE Transactions onFuzzy Systems vol 16 no 6 pp 1503ndash1521 2008

8 Mathematical Problems in Engineering

[54] S Coupland J M Mendel and D Wu ldquoEnhanced intervalapproach for encoding words into interval type-2 fuzzy sets andconvergence of the word FOUsrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10) pp1ndash8 Barcelona Spain July 2010

[55] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[56] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[57] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[58] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[59] W Wang and X Xin ldquoDistance measure between intuitionisticfuzzy setsrdquo Pattern Recognition Letters vol 26 no 13 pp 2063ndash2069 2005

[60] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[61] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[62] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[63] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[64] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[65] Y Xu and H Wang ldquoDistance measure for linguistic decisionmakingrdquo Systems Engineering Procedia vol 1 pp 450ndash456 2011

[66] Y M Wang J B Yang and D L Xu ldquoA preference aggregationmethod through the estimation of utility intervalsrdquo Computersand Operations Research vol 32 no 8 pp 2027ndash2049 2005

[67] H Liao Z Xu and X Zeng ldquoDistance and similarity measuresfor hesitant fuzzy linguistic term sets and their application inmulti-criteria decision makingrdquo Information Sciences vol 271pp 125ndash142 2014

[68] B Zhu Z Xu andMXia ldquoHesitant fuzzy geometric Bonferronimeansrdquo Information Sciences vol 205 pp 72ndash85 2012

[69] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[70] X Gu Y Wang and B Yang ldquoA method for hesitant fuzzymultiple attribute decision making and its application to riskinvestmentrdquo Journal of Convergence Information Technologyvol 6 no 6 pp 282ndash287 2011

[71] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013

[72] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012

[73] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[54] S Coupland J M Mendel and D Wu ldquoEnhanced intervalapproach for encoding words into interval type-2 fuzzy sets andconvergence of the word FOUsrdquo in Proceedings of the 6th IEEEWorld Congress on Computational Intelligence (WCCI rsquo10) pp1ndash8 Barcelona Spain July 2010

[55] L Martınez and F Herrera ldquoAn overview on the 2-tuplelinguistic model for computing with words in decision makingextensions applications and challengesrdquo Information Sciencesvol 207 pp 1ndash18 2012

[56] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[57] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[58] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[59] W Wang and X Xin ldquoDistance measure between intuitionisticfuzzy setsrdquo Pattern Recognition Letters vol 26 no 13 pp 2063ndash2069 2005

[60] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[61] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[62] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011

[63] Z Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011

[64] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[65] Y Xu and H Wang ldquoDistance measure for linguistic decisionmakingrdquo Systems Engineering Procedia vol 1 pp 450ndash456 2011

[66] Y M Wang J B Yang and D L Xu ldquoA preference aggregationmethod through the estimation of utility intervalsrdquo Computersand Operations Research vol 32 no 8 pp 2027ndash2049 2005

[67] H Liao Z Xu and X Zeng ldquoDistance and similarity measuresfor hesitant fuzzy linguistic term sets and their application inmulti-criteria decision makingrdquo Information Sciences vol 271pp 125ndash142 2014

[68] B Zhu Z Xu andMXia ldquoHesitant fuzzy geometric Bonferronimeansrdquo Information Sciences vol 205 pp 72ndash85 2012

[69] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[70] X Gu Y Wang and B Yang ldquoA method for hesitant fuzzymultiple attribute decision making and its application to riskinvestmentrdquo Journal of Convergence Information Technologyvol 6 no 6 pp 282ndash287 2011

[71] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013

[72] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012

[73] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of