Research ArticleHesitant Fuzzy Soft Sets with Application in Multicriteria GroupDecision Making Problems

Jian-qiang Wang Xin-E Li and Xiao-hong Chen

School of Business Central South University Changsha 410083 China

Correspondence should be addressed to Jian-qiang Wang jqwangcsueducn

Received 15 June 2014 Revised 25 August 2014 Accepted 26 August 2014

Academic Editor Feng Feng

Copyright copy 2015 Jian-qiang Wang et al 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

Soft sets have been regarded as a useful mathematical tool to deal with uncertainty In recent years many scholars have shownan intense interest in soft sets and extended standard soft sets to intuitionistic fuzzy soft sets interval-valued fuzzy soft sets andgeneralized fuzzy soft sets In this paper hesitant fuzzy soft sets are defined by combining fuzzy soft sets with hesitant fuzzy sets Andsome operations on hesitant fuzzy soft sets based on Archimedean t-norm and Archimedean t-conorm are defined Besides fouraggregation operations such as the HFSWA HFSWG GHFSWA and GHFSWG operators are given Based on these operatorsa multicriteria group decision making approach with hesitant fuzzy soft sets is also proposed To demonstrate its accuracy andapplicability this approach is finally employed to calculate a numerical example

1 Introduction

Since the fuzzy set (FS) was proposed by Zadeh in 1965 [1] ithas been widely studied developed and successfully appliedin various fields such as multicriteria decision making(MCDM) [2 3] fuzzy logic and approximate reasoning [4]and pattern recognition [5] In real MCDM cases due tothe fuzziness and uncertainty of decision making problemsthe criteriarsquos weights and evaluation values of alternatives canbe inaccurate uncertain or incomplete For the problemslike those FSs especially fuzzy numbers can provide goodsolutions However in FSs the membership degree of theelement is represented by a single value between zero andone and a major drawback of FSs is that single values cannotconvey information precisely

In practice the information regarding alternatives whenreferring to a fuzzy concept may be incomplete that is thesum of the membership and nonmembership degree of anelement in the universe can be less than one The FS failswhen it comes to managing the insufficient understandingofmembership degreesThus Atanassovrsquos intuitionistic fuzzysets (IFSs) interval-valued intuitionistic fuzzy sets (IVIFSs)

and trapezoidal or triangular intuitionistic fuzzy sets as theextensions of Zadehrsquos FSs were introduced [6ndash11] IFSs andIVIFSs have been widely applied in solvingMCDMproblems[10 12ndash14]

However in some cases the membership degree of anelement is neither a single value nor an interval but a set ofpossible values To manage such situations where decision-makers are hesitant in expressing their preferences overalternatives hesitant fuzzy sets (HFSs) another extensionof traditional FSs provide a useful reference HFSs are firstintroduced by Torra [15 16] and permit the membershipdegree of an element to be a set of several possible valuesbetween 0 and 1 HFSs are tremendously useful in handlingthe situations where people have hesitancy in providing theirpreferences over objects in a decision making process Theaggregation operators of HFSs were studied and appliedto MCDM problems in [17ndash20] Besides Wang et al [21]provided an outranking approach with HFSs to solveMCDMproblems Yu et al [22] and Chen et al [23] discussed thecorrelation coefficients of HFSs and their applications toclustering analysis Xu and Xia [24 25] discussed the distanceand correlation measures for HFSs

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 806983 14 pageshttpdxdoiorg1011552015806983

2 The Scientific World Journal

However in somepractical cases FSs and their extensionsfailed to model uncertain data being of various types becauseof their inadequacy as a parameterization tool To overcomethis difficulty Molodtsov [26] proposed soft sets (SSs) whichwere considered as a useful mathematical tool for dealingwith uncertainties which is free from the difficulties affectingthe exiting methods After that many scholars have shownan intense interest in this Maji et al [27] gave a theoreticalstudy of SSs They [28] also described the application of SSsin a decision making problem In addition the operationsof SSs were extended in [29ndash31] Min [32] proposed thesimilarity measures between SSs Aktas and Cagman [33]gave a definition of soft groups and discussed the basicproperties Acar et al [34] introduced soft rings Gong et al[35] proposed bijective SSs and the corresponding operationsFurthermore the relations and functions of SSs were studiedby Babitha and Sunil [36] Ali et al studied the algebraicstructures of SSs [37] and also discussed the idea of reductionof parameters in case of SSs [38] Cagman and Enginoglu [39]defined the products of SSs and uni-int decision function andthen constructed a uni-int decision making method Feng etal [40] improved Cagman and Enginoglursquos uni-int decisionmaking method based on choice value being in the form ofSSs

But situations in real world may be complex because ofthe fuzzy nature of parameters To deal with such situationMaji et al [41] extended classical SSs to fuzzy soft sets(FSSs) Borah et al [42] discussed some operations of FSSsGuan et al [43] gave a new order relation of FSSs and Fenget al [44] studied the decomposition of FSSs with finitevalue spaces Similar to FS theory several new extensionalconcepts based on FSSs are given For exampleMaji et al [45]introduced intuitionistic fuzzy soft sets (IFSSs) by integratingSSs with IFSs To overcome the difficulties of representationof parameterrsquos vagueness Xu et al [46] introduced vague softset (VSSs) and Zhou and Li [47] defined generalized vaguesoft sets (GVSSs) By combining IVFSs and SSs Yang et al[48] proposed interval-valued fuzzy soft sets (IVFSSs) Maet al [49] analyzed the parameter reduction of IVFSSs andJiang et al [50] studied the entropy of IFSSs and IVFSSs Inaddition Majumdar and Samanta [51] defined generalizedfuzzy soft sets (GFSSs) Moreover FSSs have been alsosuccessfully applied in MCDM problems in recent years RoyandMaji [52] gave an approach to decisionmaking problemsBy means of level soft sets Feng et al [53] presented anadjustable approach to FSSs based decision making Konget al [54] presented a decision making algorithm of FSSsbased on grey theory Mitra Basu et al [55] proposed abalanced solution of FSSs in medical science Xiao et al [56]integrated the fuzzy cognitive map and FSSs for solving thesupplier selection problem In [57 58] the approaches toMCDM based on IFSSs are given Jiang et al [59] presentedan adjustable approach to IFSSs-based decision making byusing level soft sets of IFSSs To deal with the problems ofsubjective evaluation and uncertain knowledge Xiao et al[60] proposed an evaluation method based on GFSSs andits application in medical diagnosis problem They also [61]extended classical SSs to trapezoidal fuzzy soft sets (TFSSs)and applied them to MCDM problems Zhang et al [62]

applied generalized TFSSs to medical diagnosis Zhang [63]presented a rough set approach to IFSSs-based decisionmaking

IFSSs [45] VSSs [46] IVFSSs [48] and GFSSs [51] areall proposed to deal with uncertainties by taking advantagesof SSs In those extensions of SSs the value of membershipis either a single value or an interval But in fact themembership degree may be a set of possible values in a SSso the purpose of this paper is to deal with this situationby combining HFSs with FSSs To do this a new kind ofSSs hesitant fuzzy soft sets (HFSSs) can be defined HFSSscan represent various different preferences from differentdecision-makers and avoid overlooking any subjective inten-tions of decision-makers Babitha and John [64] introducedHFSSs and analyzed some basis operations However theMCDMmethod proposed by Babitha and John [64] was notpersuadable In fact HFSSs are more suitable for multiplecriteria group decision making (MCGDM) problems In thispaper a further study of HFSSs and their application inMCGDM problems are given

The rest of this paper is organized as follows In Section 2some basic concepts of t-norm t-conorm SSs FSSs andHFSs are briefly reviewed In Section 3 the concept ofHFSSs and the corresponding operations are introduced InSection 4 some aggregation operators of HFSSs are givenBased on hesitant fuzzy soft numbers (HFSNs) a MCGDMapproach is proposed in Section 5 An illustrative exampleis given in Section 6 and the conclusions are provided inSection 7

2 Preliminaries

In this section some basic concepts of t-norm t-conorm SSsFSSs and HFSs are reviewed

21 T-Norm and T-Conorm

Definition 1 (see [65 66]) A function 119879 [0 1] times [0 1] rarr

[0 1] is called a t-norm if the following conditions are true

(1) for all 119909 isin [0 1] 119879(1 119909) = 119909(2) for all 119909 119910 isin [0 1] 119879(119909 119910) = 119879(119910 119909)(3) for all 119909 119910 119911 isin [0 1] 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)(4) if 119909 le 119909

1015840 119910 le 1199101015840 then 119879(119909 119910) = 119879(119909

1015840 1199101015840)

Definition 2 (see [65 66]) A function 119878 [0 1] times [0 1] rarr

[0 1] is called a t-conorm if the following four conditions aretrue

(1) for all 119909 isin [0 1] 119878(0 119909) = 119909(2) for all 119909 119910 isin [0 1] 119878(119909 119910) = 119878(119910 119909)(3) for all 119909 119910 119911 isin [0 1] 119878(119909 119878(119910 119911)) = 119878(119878(119909 119910) 119911)(4) if 119909 le 119909

1015840 119910 le 1199101015840 then 119878(119909 119910) = 119878(119909

1015840 1199101015840)

Definition 3 (see [65 66]) A t-norm function 119879(119909 119910) iscalledArchimedean t-norm if it is continuous and for all 119909 isin

(0 1) 119879(119909 119909) lt 119909 If for all 119909 119910 isin (0 1) 119879(119909 119910) is strictly

The Scientific World Journal 3

increasing 119879(119909 119910) can be called strictly Archimedean t-norm A t-conorm function 119878(119909 119910) is called Archimedean t-conorm if it is continuous and for all 119909 isin (0 1) 119878(119909 119909) lt 119909If for all 119909 119910 isin (0 1) 119878(119909 119910) is strictly increasing 119878(119909 119910) canbe called strictly Archimedean t-conorm

It is well known [67] that a strictly Archimedean t-normis generated by its additive generator 119896 as119879(119909 119910) = 119896

minus1(119896(119909)+

119896(119910)) and 119896 is a strictly decreasing function [0 1] rarr [0infin)

such that 119896(1) = 0 Let 119897(119905) = 119896(1 minus 119905) and then Archimedeant-conorm can be expressed as 119878(119909 119910) = 119897

minus1(119897(119909) + 119897(119910))

If we use specific forms to represent 119896 then some t-normsand t-conorms can be obtained

(1) Assuming 119896(119905) = minus log 119905 then 119897(119905) = minus log(1 minus 119905)119896minus1(119905) = 119890

minus119905 and 119897minus1(119905) = 1 minus 119890

minus119905 Algebraic t-conormand t-norm [68] can be obtained 119878119860(119909 119910) = 119909 + 119910 minus

119909119910 and 119879119860(119909 119910) = 119909119910

(2) Assuming 119896(119905) = log((2 minus 119905)119905) then 119897(119905) = log((2 minus

(1 minus 119905))(1 minus 119905)) 119896minus1(119905) = 2(119890119905+ 1) and 119897

minus1(119905) = 1 minus

(2(119890119905+1)) Einstein t-conorm and t-norm [68] can be

obtained 119878119864(119909 119910) = (119909 + 119910)(1 + 119909119910) and 119879119864(119909 119910) =119909119910(1 + (1 minus 119909)(1 minus 119910))

22 Soft Sets and Fuzzy Soft Sets In this subsection thedefinitions of SSs and FSSs are introduced

Definition 4 (see [26]) Let 119880 be an initial universe and let 119864be a set of parameters A pair (119865 119864) is called soft set (SS) over119880 where 119865 is a mapping of 119864 into the set of all subsets of 119880

In otherwords any SS is a parameterized family of subsetsof the set 119880 For all 119890 isin 119864 119865(119890) may be considered the setof elements of the sets (119865 119864) or the set of 119890-approximateelements of the SS To illustrate this idea let us consider thefollowing example

Example 5 Suppose that 119880 = ℎ1 ℎ2 ℎ3 ℎ4 is a set of

houses and 119864 = 1198901 1198902 1198903 is a set of parameters which

stand for being beautiful being cheap and being in thegreen surroundings respectively Consider themapping fromparameter set 119864 to the set of all subsets of 119880 Then SS (119865 119864)can describe an ldquoattractiverdquo house that Mr X is going to buy

119865 (1198901) = ℎ

2 ℎ4 119865 (119890

2) = ℎ

1 ℎ3

119865 (1198903) = ℎ

3 ℎ4

(1)

Thus we can view the SS (119865 119864) as a collection ofapproximations as follows (119865 119864) = beautiful houses =ℎ2 ℎ4 cheap houses = ℎ

1 ℎ3 in the green surroundings

houses = ℎ3 ℎ4

For the purpose of storing a SS in a computer we couldrepresent the SS of Example 5 in Table 1

Definition 6 (see [41]) Let 119865(119880) be the set of all fuzzy subsetsof119880 and then a pair (119865 119864) is called a fuzzy soft set (FSS) over119865(119880) where 119865 is a mapping denoted by

119865 119864 997888rarr 119865 (119880) (2)

Table 1 The tabular representation of the SS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

0 1 0ℎ2

1 0 0ℎ3

0 1 1ℎ4

1 0 1

Example 7 Consider Example 5 If Mr X thinks ℎ1is a little

expensive and this fuzzy information cannot be expressedonly by two crisp numbers that is 0 and 1 a membershipdegree can be used instead which is associated with eachelement and represented by a real number in the interval[0 1]Then FSS (119865 119864) can describe the ldquoattractiverdquo house thatMr X is going to buy under the fuzzy information

119865 (1198901) =

ℎ1

02ℎ2

07ℎ3

01ℎ4

07

119865 (1198902) =

ℎ1

08ℎ2

03ℎ3

07ℎ4

01

119865 (1198903) =

ℎ1

01ℎ2

02ℎ3

09ℎ4

08

(3)

Similarly the representation of the FSS of Example 7 canbe shown in Table 2

23 Hesitant Fuzzy Sets

Definition 8 (see [15]) Let 119883 be a universal set and then ahesitant fuzzy set (HFS) on119883 is defined in terms of a functionℎ that when applied to 119883 returns a finite subset of [0 1] AHFS can be represented by

119864 = ⟨119909 ℎ119864(119909)⟩ | 119909 isin 119883 (4)

where ℎ119864(119909) is a set of values in [0 1] denoting the possible

membership degrees of the element 119909 isin 119883 to the set E ℎ119864(119909)

is called a hesitant fuzzy element (HFE) [17] and 119867 is theset of all HFEs In particular if 119883 has only one element wecall 119864 a hesitant fuzzy number (HFN) briefly denoted by 119864 =

ℎ119864(119909) The set of all hesitate fuzzy numbers is represented

as HFNs

Torra [15] defined some operations onHFNs andXia andXu [17] defined some new operations on HFNs and also thescore functions

Definition 9 (see [15]) Let ℎ1 ℎ2 ℎ isin 119867 and three operations

are defined as follows

(1) ℎ119888 = cup120574isinℎ

1 minus 120574(2) ℎ1cup ℎ2= cup1205741isinℎ11205742isinℎ2

max1205741 1205742

(3) ℎ1cap ℎ2= cap1205741isinℎ11205742isinℎ2

min1205741 1205742

The arithmetical operations of HFNs based on Archime-dean t-norm and Archimedean t-conorm are defined asfollows

4 The Scientific World Journal

Table 2 The tabular representation of the FSS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

02 08 01ℎ2

07 03 02ℎ3

01 07 09ℎ4

07 01 08

Definition 10 (see [17]) Let ℎ1= cup1205741isinℎ1

1205741 ℎ2= cup1205742isinℎ2

1205742

and ℎ = cup120574isinℎ

120574 be three HFNs and 120582 ge 0 Four operationsare defined as follows

(1) ℎ120582 = cup120574isinℎ

119896minus1(120582119896(120574))

(2) 120582ℎ = cup120574isinℎ

119897minus1(120582119897(120574))

(3) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

119897minus1(119897(1205741) + 119897(120574

2))

(4) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

119896minus1(119896(1205741) + 119896(120574

2))

In particular if 119896(119905) = minus log 119905 then(5) ℎ120582 = cup

120574isinℎ120574120582

(6) 120582ℎ = cup120574isinℎ

1 minus (1 minus 120574)120582

(7) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

1205741+ 1205742minus 12057411205742

(8) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

12057411205742

If 119896(119905) = log((2 minus 119905)119905) then

(9) ℎ120582 = cup120574isinℎ

2120574120582((2 minus 120574)

120582+ 120574120582)

(10) 120582ℎ = cup120574isinℎ

((1 + 120574)120582minus (1 minus 120574)

120582)((1 + 120574)

120582+ (1 minus 120574)

120582)

(11) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

(1205741+ 1205742)(1 + 120574

11205742)

(12) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

12057411205742(1 + (1 minus 120574

1)(1 minus 120574

2))

Definition 11 (see [17]) For ℎ isin HFN 119904(ℎ) = (1ℎ)sum120574isinℎ

120574

is called the score function of ℎ where ℎ is the number ofelements in ℎ For two HFNs and ℎ

2 if 119904(ℎ

1) gt 119904(ℎ

2) then

ℎ1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2

It is clear that Definition 11 does not consider the situationwhere two HFNs ℎ

1and ℎ

2have the same score but their

deviation degreesmay be differentThe deviation degree of allelements with respect to the average value in a HFN reflectshow elements are consistent with each other that is whetherthey have a higher consistency or not To better represent thisissue Chen et al [69] defined the deviation degree as follows

Definition 12 (see [69]) For ℎ isin HFN 120590(ℎ) = [(1ℎ)sum120574isinℎ

(120574minus119904(ℎ))2]12 is defined as the variance of ℎ where 119904(ℎ) is

the score function of ℎ and 120590(ℎ) denotes the deviation degreeof ℎ

Definition 13 (see [69]) Let ℎ1and ℎ2be twoHFNs if 119904(ℎ

1) gt

119904(ℎ2) then ℎ

1gt ℎ2

If 119904(ℎ1) = 119904(ℎ

2) then

(1) if 120590(ℎ1) gt 120590(ℎ

2) then ℎ

1lt ℎ2

(2) if 120590(ℎ1) lt 120590(ℎ

2) then ℎ

1gt ℎ2

(3) if 120590(ℎ1) = 120590(ℎ

2) then ℎ

1= ℎ2

3 Hesitant Fuzzy Soft Sets andTheir Operations

In this section aHFSS is defined by combiningHFSs and SSssome operations on HFSSs based on Archimedean t-normand Archimedean t-conorm are also given

31 Definition of Hesitant Fuzzy Soft Sets HFSs and SSs areintegrated as mentioned in Section 2 The concept of HFSSsis defined as follows

Definition 14 Let 119880 be an universe let 119864 be a set ofparameters and let119865(119880) be the set of all hesitant fuzzy subsetsof 119880 A pair (119865 119864) is called a HFSS over 119880 where 119865 is amapping denoted by

119865 119864 997888rarr 119865 (119880) (5)

AHFSS is a parameterized family of hesitant fuzzy subsetsof119880 that is119865(119880) For all 120576 isin 119864 119865(120576) is referred to as the set of120576-approximate elements of the HFSS (119865 119864) It can be writtenas

119865 (120576) = ⟨119909 120583119865(120576)(119909)

⟩ | 119909 isin 119880 (6)

Since HFE can represent the situation in which differentmembership functions are considered possible [15] 120583

119865(120576)(119909)

is a set of several possible values which is the hesitantfuzzy membership degree In particular if 119865(120576) has only oneelement 119865(120576) can be called a hesitant fuzzy soft number(HSSN) For convenience a HSSN is denoted by 119865(120576) =

⟨119909 120583119865(120576)(119909)

⟩

Example 15 Consider Example 7 Mr X found it was hardto give a single value to express his opinion about thehouses with respect to different criteria For example Mr Xthinks that the degree of house ℎ

1satisfies that criterion 119890

1

ldquobeautifulrdquo is 03 or 02 Then a HFSS (119865 119864) can be used todescribe the ldquoattractiverdquo house that Mr X is going to buy

119865 (1198901) =

ℎ1

03 02

ℎ2

08 07

ℎ3

02 01

ℎ4

07

119865 (1198902) =

ℎ1

09 08

ℎ2

03

ℎ3

07 06

ℎ4

01

119865 (1198903) =

ℎ1

01

ℎ2

03 02

ℎ3

09 07

ℎ4

08

(7)

For the purpose of storing a HFSS in a computer theHFSS of Example 15 is shown in Table 3

32Operations onHesitant Fuzzy Soft Sets In this subsectionsome operations on HFSNs are defined

Definition 16 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 and119865(119890119895) = ℎ

119901120583119895119901

| 119901 = 1 2 119898 be two HFSNs and 120582 gt 0Then the following can be defined

(1) (119865(119890119894))119888= ℎ119901cup120574119894119901isin120583119894119901

1 minus 120574119894119901 | 119901 = 1 2 119898

(2) 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1(120582119897(120574119894119901)) | 119901 = 1 2 119898

The Scientific World Journal 5

Table 3 The tabular representation of the HFSS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

(3) 119865(119890119894)120582= ℎ119901cup120574119894119901isin120583119894119901

119896minus1(119896119897(120574119894119901)) | 119901 = 1 2 119898

(4) 119865(119890119894) oplus 119865(119890

119895) = ℎ

119901cup120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

119895119901)) |

119901 = 1 2 119898

(5) 119865(119890119894) otimes 119865(119890

119895) = ℎ

119901cup120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119896minus1(119896(120574119894119901) +

119896(120574119895119901)) | 119901 = 1 2 119898

Theorem 17 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 and119865(119890119895) = ℎ

119901120583119895119901

| 119901 = 1 2 119898 be two HFSNs and120582 1205821 1205822gt 0 Then the following are true

(1) 119865(119890119894) oplus 119865(119890

119895) = 119865(119890

119895) oplus 119865(119890

119894)

(2) 119865(119890119894) otimes 119865(119890

119895) = 119865(119890

119895) otimes 119865(119890

119894)

(3) 120582(119865(119890119894) oplus 119865(119890

119895)) = 120582119865(119890

119894) oplus 120582119865(119890

119895)

(4) (119865(119890119894) otimes 119865(119890

119895))120582= 119865(119890119894)120582otimes 119865(119890119895)120582

(5) 1205821F(ei) oplus 120582

2F(ei) = (120582

1+ 1205822)F(ei)

(6) (119865(119890119894))1205821 otimes (119865(119890

119894))1205822 = (119865(119890

119894))1205821+1205822

Proof InDefinition 16 it is easy to prove that (1) and (2)holdand thus let us prove (3) Consider

120582 (119865 (119890119894) oplus 119865 (119890

119895))

= 120582

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (119897 (120574119894119901) + 119897 (120574

119895119901))

| 119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582119897 (119897minus1 (119897 (120574119894119901) + 119897 (120574

119895119901))))

|

119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582 (119897 (120574119894119901) + 119897 (120574

119895119901)))

| 119901 isin 119875

120582119865 (119890119894) oplus 120582119865 (119890

119895)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (120574119894119901))

| 119901 isin 119875

oplus

ℎ119901

⋃120574119895119901isin120583119895119901

119897minus1 (120582119897 (120574119895119901))

| 119901 isin 119875

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1(119897 (119897minus1(120582119897 (120574

119894119901)))

+ 119897 (119897minus1(120582119897 (120574

119895119901)))))

minus1

|

119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582 (119897 (120574119894119901) + 119897 (120574

119895119901)))

| 119901 isin 119875

= 120582 (119865 (119890119894) oplus 119865 (119890

119895))

(8)

where 119875 = 1 2 119898Similarly (4)ndash(6) of Definition 16 can be proved

4 Aggregation Operators of Hesitant FuzzySoft Sets

In this section some aggregation operators are defined

Definition 18 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWA

operator can be called hesitant fuzzy soft weighted averagingoperator and defined as

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

120596119894119865 (119890119894) (9)

Theorem 19 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using the HFSWA operator is still a HFSN and

119867119865119878119882119860(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

119899

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(10)

Proof Use the mathematical introduction of 119899 to completethis proof

For 119899 = 2 we have

HFSWA (119865 (1198901) 119865 (119890

2))

=

2

⨁

119894=1

120596119894119865 (119890119894) = 1205961119865 (1198901) oplus 1205962119865 (1198902)

6 The Scientific World Journal

=

ℎ119901

⋃1205741119901isin12058311199011205742119901isin1205832119901

119897minus1 (1205961119897 (1205741119901) + 1205962119897 (1205742119901))

|

119901 = 1 2 119898

(11)

Suppose (10) holds for 119899 = 119896 that is

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896))

=

119896

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896) 119865 (119890

119896+1))

=

119896

⨁

119894=1

120596119894119865 (119890119894) oplus 120596119894+1

119865 (119890119894+1

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574119896+1119901isin120583119896+1119901

119897minus1 (120582119897 (120574119896+1119901

))

| 119901 = 1 2 119898

=

ℎ119901( ⋃

120574119894119901isin120583119894119901

119897minus1(119897(119897minus1(

119896

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897 (119897minus1(120596119896+1

119897 (120574119896+1119901

)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901) + 120596119896+1

119897 (120574119896+1119901

))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896+1

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(12)

Now (10) holds for 119899 = 119896+1 and thus (10) holds for all 119899Subsequently some desirable properties of the HFSWA

operator are investigated

Property 1 If all 119865(119890119894) (119894 = 1 2 119899) are equal that is

119865(119890119894) = 119865(119890) = ℎ

119901120583119894119901| 119901 = 1 2 119898 for all 119894 then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = 119865 (119890) (13)

Proof Let 119865(119890119894) = 119865(119890) and then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= HFSWA (119865 (119890) 119865 (119890) 119865 (119890)) =

119899

⨁

119894=1

120596119894119865 (119890)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

120583119894119901

| 119901 = 1 2 119898 (119894 = 1 2 119899) = 119865 (119890)

(14)

Property 2 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 119865(120576) =

ℎ119901120583120576119901

| 119901 = 1 2 119898 is a hesitant fuzzy soft element(HFSE) then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(15)

Proof Since 119865(119890119895) oplus 119865(120576) = ℎ

119901cup120574119895119901isin120583119895119901120574120576119901isin120583120576119901

119897minus1(119897(120574119895119901) +

119897(120574120576119901)) | 119901 = 1 2 119898 then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119901)))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

120583120576119901

| 119901 = 1 2 119898

The Scientific World Journal 7

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119901isin120583120576119901

119897minus1(119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901))))

+ 119897 (120574120576119901))

minus1

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

(16)

Therefore HFSWA (119865(1198901) oplus 119865(120576) 119865(119890

2) oplus 119865(120576) 119865(119890

119899) oplus

119865(120576)) =HFSWA (119865(1198901) 119865(1198902) 119865(119890

119899)) oplus 119865(120576)

Property 3 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0

then

HFSWA (120582119865 (1198901) 120582119865 (119890

2) 120582119865 (119890

119899))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

(17)

Proof According to Definition 16 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1

(120582119897(120574119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (120582119897 (120574

119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894(120582119897 (120574

119894119901)))

| 119901 = 1 2 119898

120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582 (sum119899

119894=1120596119894119897 (120574119894119901)))

| 119901 = 1 2 119898

(18)

Therefore the proof of Property 3 is completed

According to Properties 2 and 3 Property 4 can be given

Property 4 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be all the elements of the HFSS (119865 119864) and let120596 = (120596

1 1205962 120596

119899) be the weight vector of 119865(119890

119894) such that

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0 and

119865(120576) = ℎ119901120583120576119901| 119901 = 1 2 119898 is a HFSE then

HFSWA (120582119865 (1198901) oplus 119865 (120576) 120582119865 (119890

2) oplus 119865 (120576)

120582119865 (119890119899) oplus 119865 (120576))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(19)

Property 5 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 1198641) let 119865(120576

119894) =

ℎ119901120583120576119894119901| 119901 = 1 2 119898 (119894 = 1 2 119899) be the elements of

HFSS (119865 1198642) and let120596 = (120596

1 1205962 120596

119899) be theweight vector

of them satisfying 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2)

119865 (119890119899) oplus 119865 (120576

119899))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

(20)

Proof According to Definition 16 119865(119890119894) oplus 119865(120576

119894) = ℎ

119901

cup120574119894119901isin120583119894119901120574120576119894119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

120576119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2) 119865 (119890

119899) oplus 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574120576119894119901))

| 119901 = 1 2 119898

8 The Scientific World Journal

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1

times (119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897(119897minus1

times(

119899

sum

119894=1

120596119894119897 (120574120576119894119901)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + sum119899

119894=1120596119894119897 (120574120576119894119901))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

(21)

Thus the proof of Property 5 is completed

Definition 20 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWG

operator can be called hesitant fuzzy soft weighted geometricoperator and defined as

HFSWG (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

(22)

Theorem 21 Let 119865(119890119894) = ℎ

119895119901119895| 119895 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using HFSWG operator is a HFSN and

119867119865119878119882119866(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 (sum119899

119894=1120596119894119896 (120574119894119901))

| 119901 = 1 2 119898

(23)Similarly it is clear that theHFSWGoperator also satisfies

the properties that HFSWA operator has and the relativedetails are omitted here

Definition 22 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

sum119899

119894=1120596119894= 1 Then the GHFSWA operator can be called the

generalized hesitant fuzzy soft weighted averaging operatorand defined as

GHFSWA120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

(24)

In particular if 120582 = 1 then the GHFSWA operator isreduced to the HFSWA operator

Theorem 23 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWA operator is aHFSN and

119866119867119865119878119882119860120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119901))))))

| 119901 = 1 2 119898

(25)

Definition 24 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

The Scientific World Journal 9

sum119899

119894=1120596119894= 1 Then the GHFSWG operator can be called the

generalized hesitant fuzzy soft weighted geometric operatorand defined as

GHFSWG120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

(26)

In particular if 120582 = 1 then the GHFSWG operator isreduced to the HFSWG operator

Theorem 25 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWG operator is aHFSN and

119866119867119865119878119882119866120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119901))))))

|

119901 = 1 2 119898

(27)

Similarly it is clear that the GHFSWA and GHFSWGoperators possess the same properties that the HFSWAoperator has

Example 26 Suppose 119865(1198901) = ℎ

103 05 ℎ

204 06

and 119865(1198902) = ℎ

104 05 ℎ

208 09 are the elements of

the HFSS (119865 119864) and 120596 = (03 07) is the weight vector of119865(119890119894) (119894 = 1 2) which indicate the corresponding importance

degrees

Using Theorems 19 and 21 if we assign 119896(119905) = minus log(119905)then

HFSWA (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

1 minus prod2

119894=1(1 minus 120574

119894119901)120596119894

| 119901 = 1 2

= ℎ1

037 045 043 050

ℎ2

072 083 075 085

HFSWG (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

prod2

119894=1120574120596119894

119894119901

| 119901 = 1 2

= ℎ1

037 043 043 050

ℎ2

065 071 073 080

(28)

5 Multicriteria Group Decision MakingApproach with HFSSs

Consider a MCGDM with hesitant fuzzy soft informationand assume there are n alternatives119880 = ℎ

1 ℎ2 ℎ

119899 andm

criteria 119864 = 1198901 1198902 119890

119898 and the weight vector of criteria

is 120596 = (1205961 1205962 120596

119898) where 120596

119895ge 0 (119895 = 1 2 119898)

and sum119898

119895=1120596119895= 1 Suppose that there are 119896 decision-makers

119863 = 1198891 1198892 119889

119896 whose corresponding weight vector

is 120578 = (1205781 1205782 120578

119896) Then a MCGDM problem can be

concisely expressed by 119896HFSSs as follows

119865119896(119890119894) =

ℎ119895

120583119894119895

| 119895 = 1 2 119899 (119894 = 1 2 119898) (29)

To obtain the best alternative the steps of the proposedMCGDM approach with HFSSs are given as follows

Step 1 (normalize the decision information) For benefit-typecriteria no further action need be taken However for cost-type criteria the negation operator given in Definition 16need be used The normalized information can be denotedas follows

119865 (119890119894) =

ℎ119895

120583119894119895

119895 isin 119861119879

ℎ119895

cup120574119894119895isin120583119894119895

1 minus 120574119894119895

119895 isin 119862119879

(119894 = 1 2 119898 119895 = 1 2 119899)

(30)

Here 119861119879is the set of benefit-type criteria and119862

119879is the set

of cost-type criteria

Step 2 (aggregate the HFSEs of each decision-maker) Calcu-late the comprehensive evaluation values of each alternativefor each decision-maker by using the GHFSWAor GHFSWGoperator

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

The Scientific World Journal 3

increasing 119879(119909 119910) can be called strictly Archimedean t-norm A t-conorm function 119878(119909 119910) is called Archimedean t-conorm if it is continuous and for all 119909 isin (0 1) 119878(119909 119909) lt 119909If for all 119909 119910 isin (0 1) 119878(119909 119910) is strictly increasing 119878(119909 119910) canbe called strictly Archimedean t-conorm

It is well known [67] that a strictly Archimedean t-normis generated by its additive generator 119896 as119879(119909 119910) = 119896

minus1(119896(119909)+

119896(119910)) and 119896 is a strictly decreasing function [0 1] rarr [0infin)

such that 119896(1) = 0 Let 119897(119905) = 119896(1 minus 119905) and then Archimedeant-conorm can be expressed as 119878(119909 119910) = 119897

minus1(119897(119909) + 119897(119910))

If we use specific forms to represent 119896 then some t-normsand t-conorms can be obtained

(1) Assuming 119896(119905) = minus log 119905 then 119897(119905) = minus log(1 minus 119905)119896minus1(119905) = 119890

minus119905 and 119897minus1(119905) = 1 minus 119890

minus119905 Algebraic t-conormand t-norm [68] can be obtained 119878119860(119909 119910) = 119909 + 119910 minus

119909119910 and 119879119860(119909 119910) = 119909119910

(2) Assuming 119896(119905) = log((2 minus 119905)119905) then 119897(119905) = log((2 minus

(1 minus 119905))(1 minus 119905)) 119896minus1(119905) = 2(119890119905+ 1) and 119897

minus1(119905) = 1 minus

(2(119890119905+1)) Einstein t-conorm and t-norm [68] can be

obtained 119878119864(119909 119910) = (119909 + 119910)(1 + 119909119910) and 119879119864(119909 119910) =119909119910(1 + (1 minus 119909)(1 minus 119910))

22 Soft Sets and Fuzzy Soft Sets In this subsection thedefinitions of SSs and FSSs are introduced

Definition 4 (see [26]) Let 119880 be an initial universe and let 119864be a set of parameters A pair (119865 119864) is called soft set (SS) over119880 where 119865 is a mapping of 119864 into the set of all subsets of 119880

In otherwords any SS is a parameterized family of subsetsof the set 119880 For all 119890 isin 119864 119865(119890) may be considered the setof elements of the sets (119865 119864) or the set of 119890-approximateelements of the SS To illustrate this idea let us consider thefollowing example

Example 5 Suppose that 119880 = ℎ1 ℎ2 ℎ3 ℎ4 is a set of

houses and 119864 = 1198901 1198902 1198903 is a set of parameters which

stand for being beautiful being cheap and being in thegreen surroundings respectively Consider themapping fromparameter set 119864 to the set of all subsets of 119880 Then SS (119865 119864)can describe an ldquoattractiverdquo house that Mr X is going to buy

119865 (1198901) = ℎ

2 ℎ4 119865 (119890

2) = ℎ

1 ℎ3

119865 (1198903) = ℎ

3 ℎ4

(1)

Thus we can view the SS (119865 119864) as a collection ofapproximations as follows (119865 119864) = beautiful houses =ℎ2 ℎ4 cheap houses = ℎ

1 ℎ3 in the green surroundings

houses = ℎ3 ℎ4

For the purpose of storing a SS in a computer we couldrepresent the SS of Example 5 in Table 1

Definition 6 (see [41]) Let 119865(119880) be the set of all fuzzy subsetsof119880 and then a pair (119865 119864) is called a fuzzy soft set (FSS) over119865(119880) where 119865 is a mapping denoted by

119865 119864 997888rarr 119865 (119880) (2)

Table 1 The tabular representation of the SS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

0 1 0ℎ2

1 0 0ℎ3

0 1 1ℎ4

1 0 1

Example 7 Consider Example 5 If Mr X thinks ℎ1is a little

expensive and this fuzzy information cannot be expressedonly by two crisp numbers that is 0 and 1 a membershipdegree can be used instead which is associated with eachelement and represented by a real number in the interval[0 1]Then FSS (119865 119864) can describe the ldquoattractiverdquo house thatMr X is going to buy under the fuzzy information

119865 (1198901) =

ℎ1

02ℎ2

07ℎ3

01ℎ4

07

119865 (1198902) =

ℎ1

08ℎ2

03ℎ3

07ℎ4

01

119865 (1198903) =

ℎ1

01ℎ2

02ℎ3

09ℎ4

08

(3)

Similarly the representation of the FSS of Example 7 canbe shown in Table 2

23 Hesitant Fuzzy Sets

Definition 8 (see [15]) Let 119883 be a universal set and then ahesitant fuzzy set (HFS) on119883 is defined in terms of a functionℎ that when applied to 119883 returns a finite subset of [0 1] AHFS can be represented by

119864 = ⟨119909 ℎ119864(119909)⟩ | 119909 isin 119883 (4)

where ℎ119864(119909) is a set of values in [0 1] denoting the possible

membership degrees of the element 119909 isin 119883 to the set E ℎ119864(119909)

is called a hesitant fuzzy element (HFE) [17] and 119867 is theset of all HFEs In particular if 119883 has only one element wecall 119864 a hesitant fuzzy number (HFN) briefly denoted by 119864 =

ℎ119864(119909) The set of all hesitate fuzzy numbers is represented

as HFNs

Torra [15] defined some operations onHFNs andXia andXu [17] defined some new operations on HFNs and also thescore functions

Definition 9 (see [15]) Let ℎ1 ℎ2 ℎ isin 119867 and three operations

are defined as follows

(1) ℎ119888 = cup120574isinℎ

1 minus 120574(2) ℎ1cup ℎ2= cup1205741isinℎ11205742isinℎ2

max1205741 1205742

(3) ℎ1cap ℎ2= cap1205741isinℎ11205742isinℎ2

min1205741 1205742

The arithmetical operations of HFNs based on Archime-dean t-norm and Archimedean t-conorm are defined asfollows

4 The Scientific World Journal

Table 2 The tabular representation of the FSS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

02 08 01ℎ2

07 03 02ℎ3

01 07 09ℎ4

07 01 08

Definition 10 (see [17]) Let ℎ1= cup1205741isinℎ1

1205741 ℎ2= cup1205742isinℎ2

1205742

and ℎ = cup120574isinℎ

120574 be three HFNs and 120582 ge 0 Four operationsare defined as follows

(1) ℎ120582 = cup120574isinℎ

119896minus1(120582119896(120574))

(2) 120582ℎ = cup120574isinℎ

119897minus1(120582119897(120574))

(3) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

119897minus1(119897(1205741) + 119897(120574

2))

(4) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

119896minus1(119896(1205741) + 119896(120574

2))

In particular if 119896(119905) = minus log 119905 then(5) ℎ120582 = cup

120574isinℎ120574120582

(6) 120582ℎ = cup120574isinℎ

1 minus (1 minus 120574)120582

(7) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

1205741+ 1205742minus 12057411205742

(8) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

12057411205742

If 119896(119905) = log((2 minus 119905)119905) then

(9) ℎ120582 = cup120574isinℎ

2120574120582((2 minus 120574)

120582+ 120574120582)

(10) 120582ℎ = cup120574isinℎ

((1 + 120574)120582minus (1 minus 120574)

120582)((1 + 120574)

120582+ (1 minus 120574)

120582)

(11) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

(1205741+ 1205742)(1 + 120574

11205742)

(12) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

12057411205742(1 + (1 minus 120574

1)(1 minus 120574

2))

Definition 11 (see [17]) For ℎ isin HFN 119904(ℎ) = (1ℎ)sum120574isinℎ

120574

is called the score function of ℎ where ℎ is the number ofelements in ℎ For two HFNs and ℎ

2 if 119904(ℎ

1) gt 119904(ℎ

2) then

ℎ1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2

It is clear that Definition 11 does not consider the situationwhere two HFNs ℎ

1and ℎ

2have the same score but their

deviation degreesmay be differentThe deviation degree of allelements with respect to the average value in a HFN reflectshow elements are consistent with each other that is whetherthey have a higher consistency or not To better represent thisissue Chen et al [69] defined the deviation degree as follows

Definition 12 (see [69]) For ℎ isin HFN 120590(ℎ) = [(1ℎ)sum120574isinℎ

(120574minus119904(ℎ))2]12 is defined as the variance of ℎ where 119904(ℎ) is

the score function of ℎ and 120590(ℎ) denotes the deviation degreeof ℎ

Definition 13 (see [69]) Let ℎ1and ℎ2be twoHFNs if 119904(ℎ

1) gt

119904(ℎ2) then ℎ

1gt ℎ2

If 119904(ℎ1) = 119904(ℎ

2) then

(1) if 120590(ℎ1) gt 120590(ℎ

2) then ℎ

1lt ℎ2

(2) if 120590(ℎ1) lt 120590(ℎ

2) then ℎ

1gt ℎ2

(3) if 120590(ℎ1) = 120590(ℎ

2) then ℎ

1= ℎ2

3 Hesitant Fuzzy Soft Sets andTheir Operations

In this section aHFSS is defined by combiningHFSs and SSssome operations on HFSSs based on Archimedean t-normand Archimedean t-conorm are also given

31 Definition of Hesitant Fuzzy Soft Sets HFSs and SSs areintegrated as mentioned in Section 2 The concept of HFSSsis defined as follows

Definition 14 Let 119880 be an universe let 119864 be a set ofparameters and let119865(119880) be the set of all hesitant fuzzy subsetsof 119880 A pair (119865 119864) is called a HFSS over 119880 where 119865 is amapping denoted by

119865 119864 997888rarr 119865 (119880) (5)

AHFSS is a parameterized family of hesitant fuzzy subsetsof119880 that is119865(119880) For all 120576 isin 119864 119865(120576) is referred to as the set of120576-approximate elements of the HFSS (119865 119864) It can be writtenas

119865 (120576) = ⟨119909 120583119865(120576)(119909)

⟩ | 119909 isin 119880 (6)

Since HFE can represent the situation in which differentmembership functions are considered possible [15] 120583

119865(120576)(119909)

is a set of several possible values which is the hesitantfuzzy membership degree In particular if 119865(120576) has only oneelement 119865(120576) can be called a hesitant fuzzy soft number(HSSN) For convenience a HSSN is denoted by 119865(120576) =

⟨119909 120583119865(120576)(119909)

⟩

Example 15 Consider Example 7 Mr X found it was hardto give a single value to express his opinion about thehouses with respect to different criteria For example Mr Xthinks that the degree of house ℎ

1satisfies that criterion 119890

1

ldquobeautifulrdquo is 03 or 02 Then a HFSS (119865 119864) can be used todescribe the ldquoattractiverdquo house that Mr X is going to buy

119865 (1198901) =

ℎ1

03 02

ℎ2

08 07

ℎ3

02 01

ℎ4

07

119865 (1198902) =

ℎ1

09 08

ℎ2

03

ℎ3

07 06

ℎ4

01

119865 (1198903) =

ℎ1

01

ℎ2

03 02

ℎ3

09 07

ℎ4

08

(7)

For the purpose of storing a HFSS in a computer theHFSS of Example 15 is shown in Table 3

32Operations onHesitant Fuzzy Soft Sets In this subsectionsome operations on HFSNs are defined

Definition 16 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 and119865(119890119895) = ℎ

119901120583119895119901

| 119901 = 1 2 119898 be two HFSNs and 120582 gt 0Then the following can be defined

(1) (119865(119890119894))119888= ℎ119901cup120574119894119901isin120583119894119901

1 minus 120574119894119901 | 119901 = 1 2 119898

(2) 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1(120582119897(120574119894119901)) | 119901 = 1 2 119898

The Scientific World Journal 5

Table 3 The tabular representation of the HFSS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

(3) 119865(119890119894)120582= ℎ119901cup120574119894119901isin120583119894119901

119896minus1(119896119897(120574119894119901)) | 119901 = 1 2 119898

(4) 119865(119890119894) oplus 119865(119890

119895) = ℎ

119901cup120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

119895119901)) |

119901 = 1 2 119898

(5) 119865(119890119894) otimes 119865(119890

119895) = ℎ

119901cup120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119896minus1(119896(120574119894119901) +

119896(120574119895119901)) | 119901 = 1 2 119898

Theorem 17 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 and119865(119890119895) = ℎ

119901120583119895119901

| 119901 = 1 2 119898 be two HFSNs and120582 1205821 1205822gt 0 Then the following are true

(1) 119865(119890119894) oplus 119865(119890

119895) = 119865(119890

119895) oplus 119865(119890

119894)

(2) 119865(119890119894) otimes 119865(119890

119895) = 119865(119890

119895) otimes 119865(119890

119894)

(3) 120582(119865(119890119894) oplus 119865(119890

119895)) = 120582119865(119890

119894) oplus 120582119865(119890

119895)

(4) (119865(119890119894) otimes 119865(119890

119895))120582= 119865(119890119894)120582otimes 119865(119890119895)120582

(5) 1205821F(ei) oplus 120582

2F(ei) = (120582

1+ 1205822)F(ei)

(6) (119865(119890119894))1205821 otimes (119865(119890

119894))1205822 = (119865(119890

119894))1205821+1205822

Proof InDefinition 16 it is easy to prove that (1) and (2)holdand thus let us prove (3) Consider

120582 (119865 (119890119894) oplus 119865 (119890

119895))

= 120582

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (119897 (120574119894119901) + 119897 (120574

119895119901))

| 119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582119897 (119897minus1 (119897 (120574119894119901) + 119897 (120574

119895119901))))

|

119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582 (119897 (120574119894119901) + 119897 (120574

119895119901)))

| 119901 isin 119875

120582119865 (119890119894) oplus 120582119865 (119890

119895)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (120574119894119901))

| 119901 isin 119875

oplus

ℎ119901

⋃120574119895119901isin120583119895119901

119897minus1 (120582119897 (120574119895119901))

| 119901 isin 119875

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1(119897 (119897minus1(120582119897 (120574

119894119901)))

+ 119897 (119897minus1(120582119897 (120574

119895119901)))))

minus1

|

119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582 (119897 (120574119894119901) + 119897 (120574

119895119901)))

| 119901 isin 119875

= 120582 (119865 (119890119894) oplus 119865 (119890

119895))

(8)

where 119875 = 1 2 119898Similarly (4)ndash(6) of Definition 16 can be proved

4 Aggregation Operators of Hesitant FuzzySoft Sets

In this section some aggregation operators are defined

Definition 18 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWA

operator can be called hesitant fuzzy soft weighted averagingoperator and defined as

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

120596119894119865 (119890119894) (9)

Theorem 19 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using the HFSWA operator is still a HFSN and

119867119865119878119882119860(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

119899

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(10)

Proof Use the mathematical introduction of 119899 to completethis proof

For 119899 = 2 we have

HFSWA (119865 (1198901) 119865 (119890

2))

=

2

⨁

119894=1

120596119894119865 (119890119894) = 1205961119865 (1198901) oplus 1205962119865 (1198902)

6 The Scientific World Journal

=

ℎ119901

⋃1205741119901isin12058311199011205742119901isin1205832119901

119897minus1 (1205961119897 (1205741119901) + 1205962119897 (1205742119901))

|

119901 = 1 2 119898

(11)

Suppose (10) holds for 119899 = 119896 that is

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896))

=

119896

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896) 119865 (119890

119896+1))

=

119896

⨁

119894=1

120596119894119865 (119890119894) oplus 120596119894+1

119865 (119890119894+1

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574119896+1119901isin120583119896+1119901

119897minus1 (120582119897 (120574119896+1119901

))

| 119901 = 1 2 119898

=

ℎ119901( ⋃

120574119894119901isin120583119894119901

119897minus1(119897(119897minus1(

119896

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897 (119897minus1(120596119896+1

119897 (120574119896+1119901

)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901) + 120596119896+1

119897 (120574119896+1119901

))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896+1

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(12)

Now (10) holds for 119899 = 119896+1 and thus (10) holds for all 119899Subsequently some desirable properties of the HFSWA

operator are investigated

Property 1 If all 119865(119890119894) (119894 = 1 2 119899) are equal that is

119865(119890119894) = 119865(119890) = ℎ

119901120583119894119901| 119901 = 1 2 119898 for all 119894 then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = 119865 (119890) (13)

Proof Let 119865(119890119894) = 119865(119890) and then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= HFSWA (119865 (119890) 119865 (119890) 119865 (119890)) =

119899

⨁

119894=1

120596119894119865 (119890)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

120583119894119901

| 119901 = 1 2 119898 (119894 = 1 2 119899) = 119865 (119890)

(14)

Property 2 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 119865(120576) =

ℎ119901120583120576119901

| 119901 = 1 2 119898 is a hesitant fuzzy soft element(HFSE) then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(15)

Proof Since 119865(119890119895) oplus 119865(120576) = ℎ

119901cup120574119895119901isin120583119895119901120574120576119901isin120583120576119901

119897minus1(119897(120574119895119901) +

119897(120574120576119901)) | 119901 = 1 2 119898 then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119901)))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

120583120576119901

| 119901 = 1 2 119898

The Scientific World Journal 7

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119901isin120583120576119901

119897minus1(119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901))))

+ 119897 (120574120576119901))

minus1

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

(16)

Therefore HFSWA (119865(1198901) oplus 119865(120576) 119865(119890

2) oplus 119865(120576) 119865(119890

119899) oplus

119865(120576)) =HFSWA (119865(1198901) 119865(1198902) 119865(119890

119899)) oplus 119865(120576)

Property 3 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0

then

HFSWA (120582119865 (1198901) 120582119865 (119890

2) 120582119865 (119890

119899))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

(17)

Proof According to Definition 16 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1

(120582119897(120574119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (120582119897 (120574

119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894(120582119897 (120574

119894119901)))

| 119901 = 1 2 119898

120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582 (sum119899

119894=1120596119894119897 (120574119894119901)))

| 119901 = 1 2 119898

(18)

Therefore the proof of Property 3 is completed

According to Properties 2 and 3 Property 4 can be given

Property 4 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be all the elements of the HFSS (119865 119864) and let120596 = (120596

1 1205962 120596

119899) be the weight vector of 119865(119890

119894) such that

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0 and

119865(120576) = ℎ119901120583120576119901| 119901 = 1 2 119898 is a HFSE then

HFSWA (120582119865 (1198901) oplus 119865 (120576) 120582119865 (119890

2) oplus 119865 (120576)

120582119865 (119890119899) oplus 119865 (120576))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(19)

Property 5 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 1198641) let 119865(120576

119894) =

ℎ119901120583120576119894119901| 119901 = 1 2 119898 (119894 = 1 2 119899) be the elements of

HFSS (119865 1198642) and let120596 = (120596

1 1205962 120596

119899) be theweight vector

of them satisfying 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2)

119865 (119890119899) oplus 119865 (120576

119899))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

(20)

Proof According to Definition 16 119865(119890119894) oplus 119865(120576

119894) = ℎ

119901

cup120574119894119901isin120583119894119901120574120576119894119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

120576119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2) 119865 (119890

119899) oplus 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574120576119894119901))

| 119901 = 1 2 119898

8 The Scientific World Journal

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1

times (119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897(119897minus1

times(

119899

sum

119894=1

120596119894119897 (120574120576119894119901)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + sum119899

119894=1120596119894119897 (120574120576119894119901))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

(21)

Thus the proof of Property 5 is completed

Definition 20 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWG

operator can be called hesitant fuzzy soft weighted geometricoperator and defined as

HFSWG (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

(22)

Theorem 21 Let 119865(119890119894) = ℎ

119895119901119895| 119895 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using HFSWG operator is a HFSN and

119867119865119878119882119866(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 (sum119899

119894=1120596119894119896 (120574119894119901))

| 119901 = 1 2 119898

(23)Similarly it is clear that theHFSWGoperator also satisfies

the properties that HFSWA operator has and the relativedetails are omitted here

Definition 22 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

sum119899

119894=1120596119894= 1 Then the GHFSWA operator can be called the

generalized hesitant fuzzy soft weighted averaging operatorand defined as

GHFSWA120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

(24)

In particular if 120582 = 1 then the GHFSWA operator isreduced to the HFSWA operator

Theorem 23 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWA operator is aHFSN and

119866119867119865119878119882119860120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119901))))))

| 119901 = 1 2 119898

(25)

Definition 24 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

The Scientific World Journal 9

sum119899

119894=1120596119894= 1 Then the GHFSWG operator can be called the

generalized hesitant fuzzy soft weighted geometric operatorand defined as

GHFSWG120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

(26)

In particular if 120582 = 1 then the GHFSWG operator isreduced to the HFSWG operator

Theorem 25 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWG operator is aHFSN and

119866119867119865119878119882119866120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119901))))))

|

119901 = 1 2 119898

(27)

Similarly it is clear that the GHFSWA and GHFSWGoperators possess the same properties that the HFSWAoperator has

Example 26 Suppose 119865(1198901) = ℎ

103 05 ℎ

204 06

and 119865(1198902) = ℎ

104 05 ℎ

208 09 are the elements of

the HFSS (119865 119864) and 120596 = (03 07) is the weight vector of119865(119890119894) (119894 = 1 2) which indicate the corresponding importance

degrees

Using Theorems 19 and 21 if we assign 119896(119905) = minus log(119905)then

HFSWA (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

1 minus prod2

119894=1(1 minus 120574

119894119901)120596119894

| 119901 = 1 2

= ℎ1

037 045 043 050

ℎ2

072 083 075 085

HFSWG (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

prod2

119894=1120574120596119894

119894119901

| 119901 = 1 2

= ℎ1

037 043 043 050

ℎ2

065 071 073 080

(28)

5 Multicriteria Group Decision MakingApproach with HFSSs

Consider a MCGDM with hesitant fuzzy soft informationand assume there are n alternatives119880 = ℎ

1 ℎ2 ℎ

119899 andm

criteria 119864 = 1198901 1198902 119890

119898 and the weight vector of criteria

is 120596 = (1205961 1205962 120596

119898) where 120596

119895ge 0 (119895 = 1 2 119898)

and sum119898

119895=1120596119895= 1 Suppose that there are 119896 decision-makers

119863 = 1198891 1198892 119889

119896 whose corresponding weight vector

is 120578 = (1205781 1205782 120578

119896) Then a MCGDM problem can be

concisely expressed by 119896HFSSs as follows

119865119896(119890119894) =

ℎ119895

120583119894119895

| 119895 = 1 2 119899 (119894 = 1 2 119898) (29)

To obtain the best alternative the steps of the proposedMCGDM approach with HFSSs are given as follows

Step 1 (normalize the decision information) For benefit-typecriteria no further action need be taken However for cost-type criteria the negation operator given in Definition 16need be used The normalized information can be denotedas follows

119865 (119890119894) =

ℎ119895

120583119894119895

119895 isin 119861119879

ℎ119895

cup120574119894119895isin120583119894119895

1 minus 120574119894119895

119895 isin 119862119879

(119894 = 1 2 119898 119895 = 1 2 119899)

(30)

Here 119861119879is the set of benefit-type criteria and119862

119879is the set

of cost-type criteria

Step 2 (aggregate the HFSEs of each decision-maker) Calcu-late the comprehensive evaluation values of each alternativefor each decision-maker by using the GHFSWAor GHFSWGoperator

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

4 The Scientific World Journal

Table 2 The tabular representation of the FSS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

02 08 01ℎ2

07 03 02ℎ3

01 07 09ℎ4

07 01 08

Definition 10 (see [17]) Let ℎ1= cup1205741isinℎ1

1205741 ℎ2= cup1205742isinℎ2

1205742

and ℎ = cup120574isinℎ

120574 be three HFNs and 120582 ge 0 Four operationsare defined as follows

(1) ℎ120582 = cup120574isinℎ

119896minus1(120582119896(120574))

(2) 120582ℎ = cup120574isinℎ

119897minus1(120582119897(120574))

(3) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

119897minus1(119897(1205741) + 119897(120574

2))

(4) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

119896minus1(119896(1205741) + 119896(120574

2))

In particular if 119896(119905) = minus log 119905 then(5) ℎ120582 = cup

120574isinℎ120574120582

(6) 120582ℎ = cup120574isinℎ

1 minus (1 minus 120574)120582

(7) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

1205741+ 1205742minus 12057411205742

(8) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

12057411205742

If 119896(119905) = log((2 minus 119905)119905) then

(9) ℎ120582 = cup120574isinℎ

2120574120582((2 minus 120574)

120582+ 120574120582)

(10) 120582ℎ = cup120574isinℎ

((1 + 120574)120582minus (1 minus 120574)

120582)((1 + 120574)

120582+ (1 minus 120574)

120582)

(11) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

(1205741+ 1205742)(1 + 120574

11205742)

(12) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

12057411205742(1 + (1 minus 120574

1)(1 minus 120574

2))

Definition 11 (see [17]) For ℎ isin HFN 119904(ℎ) = (1ℎ)sum120574isinℎ

120574

is called the score function of ℎ where ℎ is the number ofelements in ℎ For two HFNs and ℎ

2 if 119904(ℎ

1) gt 119904(ℎ

2) then

ℎ1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2

It is clear that Definition 11 does not consider the situationwhere two HFNs ℎ

1and ℎ

2have the same score but their

deviation degreesmay be differentThe deviation degree of allelements with respect to the average value in a HFN reflectshow elements are consistent with each other that is whetherthey have a higher consistency or not To better represent thisissue Chen et al [69] defined the deviation degree as follows

Definition 12 (see [69]) For ℎ isin HFN 120590(ℎ) = [(1ℎ)sum120574isinℎ

(120574minus119904(ℎ))2]12 is defined as the variance of ℎ where 119904(ℎ) is

the score function of ℎ and 120590(ℎ) denotes the deviation degreeof ℎ

Definition 13 (see [69]) Let ℎ1and ℎ2be twoHFNs if 119904(ℎ

1) gt

119904(ℎ2) then ℎ

1gt ℎ2

If 119904(ℎ1) = 119904(ℎ

2) then

(1) if 120590(ℎ1) gt 120590(ℎ

2) then ℎ

1lt ℎ2

(2) if 120590(ℎ1) lt 120590(ℎ

2) then ℎ

1gt ℎ2

(3) if 120590(ℎ1) = 120590(ℎ

2) then ℎ

1= ℎ2

3 Hesitant Fuzzy Soft Sets andTheir Operations

In this section aHFSS is defined by combiningHFSs and SSssome operations on HFSSs based on Archimedean t-normand Archimedean t-conorm are also given

31 Definition of Hesitant Fuzzy Soft Sets HFSs and SSs areintegrated as mentioned in Section 2 The concept of HFSSsis defined as follows

Definition 14 Let 119880 be an universe let 119864 be a set ofparameters and let119865(119880) be the set of all hesitant fuzzy subsetsof 119880 A pair (119865 119864) is called a HFSS over 119880 where 119865 is amapping denoted by

119865 119864 997888rarr 119865 (119880) (5)

AHFSS is a parameterized family of hesitant fuzzy subsetsof119880 that is119865(119880) For all 120576 isin 119864 119865(120576) is referred to as the set of120576-approximate elements of the HFSS (119865 119864) It can be writtenas

119865 (120576) = ⟨119909 120583119865(120576)(119909)

⟩ | 119909 isin 119880 (6)

Since HFE can represent the situation in which differentmembership functions are considered possible [15] 120583

119865(120576)(119909)

is a set of several possible values which is the hesitantfuzzy membership degree In particular if 119865(120576) has only oneelement 119865(120576) can be called a hesitant fuzzy soft number(HSSN) For convenience a HSSN is denoted by 119865(120576) =

⟨119909 120583119865(120576)(119909)

⟩

Example 15 Consider Example 7 Mr X found it was hardto give a single value to express his opinion about thehouses with respect to different criteria For example Mr Xthinks that the degree of house ℎ

1satisfies that criterion 119890

1

ldquobeautifulrdquo is 03 or 02 Then a HFSS (119865 119864) can be used todescribe the ldquoattractiverdquo house that Mr X is going to buy

119865 (1198901) =

ℎ1

03 02

ℎ2

08 07

ℎ3

02 01

ℎ4

07

119865 (1198902) =

ℎ1

09 08

ℎ2

03

ℎ3

07 06

ℎ4

01

119865 (1198903) =

ℎ1

01

ℎ2

03 02

ℎ3

09 07

ℎ4

08

(7)

For the purpose of storing a HFSS in a computer theHFSS of Example 15 is shown in Table 3

32Operations onHesitant Fuzzy Soft Sets In this subsectionsome operations on HFSNs are defined

Definition 16 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 and119865(119890119895) = ℎ

119901120583119895119901

| 119901 = 1 2 119898 be two HFSNs and 120582 gt 0Then the following can be defined

(1) (119865(119890119894))119888= ℎ119901cup120574119894119901isin120583119894119901

1 minus 120574119894119901 | 119901 = 1 2 119898

(2) 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1(120582119897(120574119894119901)) | 119901 = 1 2 119898

The Scientific World Journal 5

Table 3 The tabular representation of the HFSS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

(3) 119865(119890119894)120582= ℎ119901cup120574119894119901isin120583119894119901

119896minus1(119896119897(120574119894119901)) | 119901 = 1 2 119898

(4) 119865(119890119894) oplus 119865(119890

119895) = ℎ

119901cup120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

119895119901)) |

119901 = 1 2 119898

(5) 119865(119890119894) otimes 119865(119890

119895) = ℎ

119901cup120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119896minus1(119896(120574119894119901) +

119896(120574119895119901)) | 119901 = 1 2 119898

Theorem 17 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 and119865(119890119895) = ℎ

119901120583119895119901

| 119901 = 1 2 119898 be two HFSNs and120582 1205821 1205822gt 0 Then the following are true

(1) 119865(119890119894) oplus 119865(119890

119895) = 119865(119890

119895) oplus 119865(119890

119894)

(2) 119865(119890119894) otimes 119865(119890

119895) = 119865(119890

119895) otimes 119865(119890

119894)

(3) 120582(119865(119890119894) oplus 119865(119890

119895)) = 120582119865(119890

119894) oplus 120582119865(119890

119895)

(4) (119865(119890119894) otimes 119865(119890

119895))120582= 119865(119890119894)120582otimes 119865(119890119895)120582

(5) 1205821F(ei) oplus 120582

2F(ei) = (120582

1+ 1205822)F(ei)

(6) (119865(119890119894))1205821 otimes (119865(119890

119894))1205822 = (119865(119890

119894))1205821+1205822

Proof InDefinition 16 it is easy to prove that (1) and (2)holdand thus let us prove (3) Consider

120582 (119865 (119890119894) oplus 119865 (119890

119895))

= 120582

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (119897 (120574119894119901) + 119897 (120574

119895119901))

| 119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582119897 (119897minus1 (119897 (120574119894119901) + 119897 (120574

119895119901))))

|

119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582 (119897 (120574119894119901) + 119897 (120574

119895119901)))

| 119901 isin 119875

120582119865 (119890119894) oplus 120582119865 (119890

119895)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (120574119894119901))

| 119901 isin 119875

oplus

ℎ119901

⋃120574119895119901isin120583119895119901

119897minus1 (120582119897 (120574119895119901))

| 119901 isin 119875

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1(119897 (119897minus1(120582119897 (120574

119894119901)))

+ 119897 (119897minus1(120582119897 (120574

119895119901)))))

minus1

|

119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582 (119897 (120574119894119901) + 119897 (120574

119895119901)))

| 119901 isin 119875

= 120582 (119865 (119890119894) oplus 119865 (119890

119895))

(8)

where 119875 = 1 2 119898Similarly (4)ndash(6) of Definition 16 can be proved

4 Aggregation Operators of Hesitant FuzzySoft Sets

In this section some aggregation operators are defined

Definition 18 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWA

operator can be called hesitant fuzzy soft weighted averagingoperator and defined as

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

120596119894119865 (119890119894) (9)

Theorem 19 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using the HFSWA operator is still a HFSN and

119867119865119878119882119860(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

119899

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(10)

Proof Use the mathematical introduction of 119899 to completethis proof

For 119899 = 2 we have

HFSWA (119865 (1198901) 119865 (119890

2))

=

2

⨁

119894=1

120596119894119865 (119890119894) = 1205961119865 (1198901) oplus 1205962119865 (1198902)

6 The Scientific World Journal

=

ℎ119901

⋃1205741119901isin12058311199011205742119901isin1205832119901

119897minus1 (1205961119897 (1205741119901) + 1205962119897 (1205742119901))

|

119901 = 1 2 119898

(11)

Suppose (10) holds for 119899 = 119896 that is

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896))

=

119896

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896) 119865 (119890

119896+1))

=

119896

⨁

119894=1

120596119894119865 (119890119894) oplus 120596119894+1

119865 (119890119894+1

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574119896+1119901isin120583119896+1119901

119897minus1 (120582119897 (120574119896+1119901

))

| 119901 = 1 2 119898

=

ℎ119901( ⋃

120574119894119901isin120583119894119901

119897minus1(119897(119897minus1(

119896

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897 (119897minus1(120596119896+1

119897 (120574119896+1119901

)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901) + 120596119896+1

119897 (120574119896+1119901

))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896+1

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(12)

Now (10) holds for 119899 = 119896+1 and thus (10) holds for all 119899Subsequently some desirable properties of the HFSWA

operator are investigated

Property 1 If all 119865(119890119894) (119894 = 1 2 119899) are equal that is

119865(119890119894) = 119865(119890) = ℎ

119901120583119894119901| 119901 = 1 2 119898 for all 119894 then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = 119865 (119890) (13)

Proof Let 119865(119890119894) = 119865(119890) and then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= HFSWA (119865 (119890) 119865 (119890) 119865 (119890)) =

119899

⨁

119894=1

120596119894119865 (119890)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

120583119894119901

| 119901 = 1 2 119898 (119894 = 1 2 119899) = 119865 (119890)

(14)

Property 2 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 119865(120576) =

ℎ119901120583120576119901

| 119901 = 1 2 119898 is a hesitant fuzzy soft element(HFSE) then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(15)

Proof Since 119865(119890119895) oplus 119865(120576) = ℎ

119901cup120574119895119901isin120583119895119901120574120576119901isin120583120576119901

119897minus1(119897(120574119895119901) +

119897(120574120576119901)) | 119901 = 1 2 119898 then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119901)))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

120583120576119901

| 119901 = 1 2 119898

The Scientific World Journal 7

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119901isin120583120576119901

119897minus1(119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901))))

+ 119897 (120574120576119901))

minus1

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

(16)

Therefore HFSWA (119865(1198901) oplus 119865(120576) 119865(119890

2) oplus 119865(120576) 119865(119890

119899) oplus

119865(120576)) =HFSWA (119865(1198901) 119865(1198902) 119865(119890

119899)) oplus 119865(120576)

Property 3 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0

then

HFSWA (120582119865 (1198901) 120582119865 (119890

2) 120582119865 (119890

119899))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

(17)

Proof According to Definition 16 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1

(120582119897(120574119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (120582119897 (120574

119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894(120582119897 (120574

119894119901)))

| 119901 = 1 2 119898

120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582 (sum119899

119894=1120596119894119897 (120574119894119901)))

| 119901 = 1 2 119898

(18)

Therefore the proof of Property 3 is completed

According to Properties 2 and 3 Property 4 can be given

Property 4 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be all the elements of the HFSS (119865 119864) and let120596 = (120596

1 1205962 120596

119899) be the weight vector of 119865(119890

119894) such that

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0 and

119865(120576) = ℎ119901120583120576119901| 119901 = 1 2 119898 is a HFSE then

HFSWA (120582119865 (1198901) oplus 119865 (120576) 120582119865 (119890

2) oplus 119865 (120576)

120582119865 (119890119899) oplus 119865 (120576))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(19)

Property 5 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 1198641) let 119865(120576

119894) =

ℎ119901120583120576119894119901| 119901 = 1 2 119898 (119894 = 1 2 119899) be the elements of

HFSS (119865 1198642) and let120596 = (120596

1 1205962 120596

119899) be theweight vector

of them satisfying 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2)

119865 (119890119899) oplus 119865 (120576

119899))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

(20)

Proof According to Definition 16 119865(119890119894) oplus 119865(120576

119894) = ℎ

119901

cup120574119894119901isin120583119894119901120574120576119894119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

120576119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2) 119865 (119890

119899) oplus 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574120576119894119901))

| 119901 = 1 2 119898

8 The Scientific World Journal

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1

times (119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897(119897minus1

times(

119899

sum

119894=1

120596119894119897 (120574120576119894119901)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + sum119899

119894=1120596119894119897 (120574120576119894119901))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

(21)

Thus the proof of Property 5 is completed

Definition 20 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWG

operator can be called hesitant fuzzy soft weighted geometricoperator and defined as

HFSWG (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

(22)

Theorem 21 Let 119865(119890119894) = ℎ

119895119901119895| 119895 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using HFSWG operator is a HFSN and

119867119865119878119882119866(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 (sum119899

119894=1120596119894119896 (120574119894119901))

| 119901 = 1 2 119898

(23)Similarly it is clear that theHFSWGoperator also satisfies

the properties that HFSWA operator has and the relativedetails are omitted here

Definition 22 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

sum119899

119894=1120596119894= 1 Then the GHFSWA operator can be called the

generalized hesitant fuzzy soft weighted averaging operatorand defined as

GHFSWA120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

(24)

In particular if 120582 = 1 then the GHFSWA operator isreduced to the HFSWA operator

Theorem 23 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWA operator is aHFSN and

119866119867119865119878119882119860120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119901))))))

| 119901 = 1 2 119898

(25)

Definition 24 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

The Scientific World Journal 9

sum119899

119894=1120596119894= 1 Then the GHFSWG operator can be called the

generalized hesitant fuzzy soft weighted geometric operatorand defined as

GHFSWG120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

(26)

In particular if 120582 = 1 then the GHFSWG operator isreduced to the HFSWG operator

Theorem 25 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWG operator is aHFSN and

119866119867119865119878119882119866120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119901))))))

|

119901 = 1 2 119898

(27)

Similarly it is clear that the GHFSWA and GHFSWGoperators possess the same properties that the HFSWAoperator has

Example 26 Suppose 119865(1198901) = ℎ

103 05 ℎ

204 06

and 119865(1198902) = ℎ

104 05 ℎ

208 09 are the elements of

the HFSS (119865 119864) and 120596 = (03 07) is the weight vector of119865(119890119894) (119894 = 1 2) which indicate the corresponding importance

degrees

Using Theorems 19 and 21 if we assign 119896(119905) = minus log(119905)then

HFSWA (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

1 minus prod2

119894=1(1 minus 120574

119894119901)120596119894

| 119901 = 1 2

= ℎ1

037 045 043 050

ℎ2

072 083 075 085

HFSWG (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

prod2

119894=1120574120596119894

119894119901

| 119901 = 1 2

= ℎ1

037 043 043 050

ℎ2

065 071 073 080

(28)

5 Multicriteria Group Decision MakingApproach with HFSSs

Consider a MCGDM with hesitant fuzzy soft informationand assume there are n alternatives119880 = ℎ

1 ℎ2 ℎ

119899 andm

criteria 119864 = 1198901 1198902 119890

119898 and the weight vector of criteria

is 120596 = (1205961 1205962 120596

119898) where 120596

119895ge 0 (119895 = 1 2 119898)

and sum119898

119895=1120596119895= 1 Suppose that there are 119896 decision-makers

119863 = 1198891 1198892 119889

119896 whose corresponding weight vector

is 120578 = (1205781 1205782 120578

119896) Then a MCGDM problem can be

concisely expressed by 119896HFSSs as follows

119865119896(119890119894) =

ℎ119895

120583119894119895

| 119895 = 1 2 119899 (119894 = 1 2 119898) (29)

To obtain the best alternative the steps of the proposedMCGDM approach with HFSSs are given as follows

Step 1 (normalize the decision information) For benefit-typecriteria no further action need be taken However for cost-type criteria the negation operator given in Definition 16need be used The normalized information can be denotedas follows

119865 (119890119894) =

ℎ119895

120583119894119895

119895 isin 119861119879

ℎ119895

cup120574119894119895isin120583119894119895

1 minus 120574119894119895

119895 isin 119862119879

(119894 = 1 2 119898 119895 = 1 2 119899)

(30)

Here 119861119879is the set of benefit-type criteria and119862

119879is the set

of cost-type criteria

Step 2 (aggregate the HFSEs of each decision-maker) Calcu-late the comprehensive evaluation values of each alternativefor each decision-maker by using the GHFSWAor GHFSWGoperator

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

The Scientific World Journal 5

Table 3 The tabular representation of the HFSS (119865 119864)

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

(3) 119865(119890119894)120582= ℎ119901cup120574119894119901isin120583119894119901

119896minus1(119896119897(120574119894119901)) | 119901 = 1 2 119898

(4) 119865(119890119894) oplus 119865(119890

119895) = ℎ

119901cup120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

119895119901)) |

119901 = 1 2 119898

(5) 119865(119890119894) otimes 119865(119890

119895) = ℎ

119901cup120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119896minus1(119896(120574119894119901) +

119896(120574119895119901)) | 119901 = 1 2 119898

Theorem 17 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 and119865(119890119895) = ℎ

119901120583119895119901

| 119901 = 1 2 119898 be two HFSNs and120582 1205821 1205822gt 0 Then the following are true

(1) 119865(119890119894) oplus 119865(119890

119895) = 119865(119890

119895) oplus 119865(119890

119894)

(2) 119865(119890119894) otimes 119865(119890

119895) = 119865(119890

119895) otimes 119865(119890

119894)

(3) 120582(119865(119890119894) oplus 119865(119890

119895)) = 120582119865(119890

119894) oplus 120582119865(119890

119895)

(4) (119865(119890119894) otimes 119865(119890

119895))120582= 119865(119890119894)120582otimes 119865(119890119895)120582

(5) 1205821F(ei) oplus 120582

2F(ei) = (120582

1+ 1205822)F(ei)

(6) (119865(119890119894))1205821 otimes (119865(119890

119894))1205822 = (119865(119890

119894))1205821+1205822

Proof InDefinition 16 it is easy to prove that (1) and (2)holdand thus let us prove (3) Consider

120582 (119865 (119890119894) oplus 119865 (119890

119895))

= 120582

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (119897 (120574119894119901) + 119897 (120574

119895119901))

| 119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582119897 (119897minus1 (119897 (120574119894119901) + 119897 (120574

119895119901))))

|

119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582 (119897 (120574119894119901) + 119897 (120574

119895119901)))

| 119901 isin 119875

120582119865 (119890119894) oplus 120582119865 (119890

119895)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (120574119894119901))

| 119901 isin 119875

oplus

ℎ119901

⋃120574119895119901isin120583119895119901

119897minus1 (120582119897 (120574119895119901))

| 119901 isin 119875

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1(119897 (119897minus1(120582119897 (120574

119894119901)))

+ 119897 (119897minus1(120582119897 (120574

119895119901)))))

minus1

|

119901 isin 119875

=

ℎ119901

⋃120574119894119901isin120583119894119901120574119895119901isin120583119895119901

119897minus1 (120582 (119897 (120574119894119901) + 119897 (120574

119895119901)))

| 119901 isin 119875

= 120582 (119865 (119890119894) oplus 119865 (119890

119895))

(8)

where 119875 = 1 2 119898Similarly (4)ndash(6) of Definition 16 can be proved

4 Aggregation Operators of Hesitant FuzzySoft Sets

In this section some aggregation operators are defined

Definition 18 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWA

operator can be called hesitant fuzzy soft weighted averagingoperator and defined as

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

120596119894119865 (119890119894) (9)

Theorem 19 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using the HFSWA operator is still a HFSN and

119867119865119878119882119860(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

119899

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(10)

Proof Use the mathematical introduction of 119899 to completethis proof

For 119899 = 2 we have

HFSWA (119865 (1198901) 119865 (119890

2))

=

2

⨁

119894=1

120596119894119865 (119890119894) = 1205961119865 (1198901) oplus 1205962119865 (1198902)

6 The Scientific World Journal

=

ℎ119901

⋃1205741119901isin12058311199011205742119901isin1205832119901

119897minus1 (1205961119897 (1205741119901) + 1205962119897 (1205742119901))

|

119901 = 1 2 119898

(11)

Suppose (10) holds for 119899 = 119896 that is

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896))

=

119896

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896) 119865 (119890

119896+1))

=

119896

⨁

119894=1

120596119894119865 (119890119894) oplus 120596119894+1

119865 (119890119894+1

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574119896+1119901isin120583119896+1119901

119897minus1 (120582119897 (120574119896+1119901

))

| 119901 = 1 2 119898

=

ℎ119901( ⋃

120574119894119901isin120583119894119901

119897minus1(119897(119897minus1(

119896

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897 (119897minus1(120596119896+1

119897 (120574119896+1119901

)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901) + 120596119896+1

119897 (120574119896+1119901

))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896+1

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(12)

Now (10) holds for 119899 = 119896+1 and thus (10) holds for all 119899Subsequently some desirable properties of the HFSWA

operator are investigated

Property 1 If all 119865(119890119894) (119894 = 1 2 119899) are equal that is

119865(119890119894) = 119865(119890) = ℎ

119901120583119894119901| 119901 = 1 2 119898 for all 119894 then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = 119865 (119890) (13)

Proof Let 119865(119890119894) = 119865(119890) and then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= HFSWA (119865 (119890) 119865 (119890) 119865 (119890)) =

119899

⨁

119894=1

120596119894119865 (119890)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

120583119894119901

| 119901 = 1 2 119898 (119894 = 1 2 119899) = 119865 (119890)

(14)

Property 2 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 119865(120576) =

ℎ119901120583120576119901

| 119901 = 1 2 119898 is a hesitant fuzzy soft element(HFSE) then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(15)

Proof Since 119865(119890119895) oplus 119865(120576) = ℎ

119901cup120574119895119901isin120583119895119901120574120576119901isin120583120576119901

119897minus1(119897(120574119895119901) +

119897(120574120576119901)) | 119901 = 1 2 119898 then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119901)))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

120583120576119901

| 119901 = 1 2 119898

The Scientific World Journal 7

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119901isin120583120576119901

119897minus1(119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901))))

+ 119897 (120574120576119901))

minus1

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

(16)

Therefore HFSWA (119865(1198901) oplus 119865(120576) 119865(119890

2) oplus 119865(120576) 119865(119890

119899) oplus

119865(120576)) =HFSWA (119865(1198901) 119865(1198902) 119865(119890

119899)) oplus 119865(120576)

Property 3 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0

then

HFSWA (120582119865 (1198901) 120582119865 (119890

2) 120582119865 (119890

119899))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

(17)

Proof According to Definition 16 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1

(120582119897(120574119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (120582119897 (120574

119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894(120582119897 (120574

119894119901)))

| 119901 = 1 2 119898

120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582 (sum119899

119894=1120596119894119897 (120574119894119901)))

| 119901 = 1 2 119898

(18)

Therefore the proof of Property 3 is completed

According to Properties 2 and 3 Property 4 can be given

Property 4 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be all the elements of the HFSS (119865 119864) and let120596 = (120596

1 1205962 120596

119899) be the weight vector of 119865(119890

119894) such that

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0 and

119865(120576) = ℎ119901120583120576119901| 119901 = 1 2 119898 is a HFSE then

HFSWA (120582119865 (1198901) oplus 119865 (120576) 120582119865 (119890

2) oplus 119865 (120576)

120582119865 (119890119899) oplus 119865 (120576))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(19)

Property 5 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 1198641) let 119865(120576

119894) =

ℎ119901120583120576119894119901| 119901 = 1 2 119898 (119894 = 1 2 119899) be the elements of

HFSS (119865 1198642) and let120596 = (120596

1 1205962 120596

119899) be theweight vector

of them satisfying 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2)

119865 (119890119899) oplus 119865 (120576

119899))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

(20)

Proof According to Definition 16 119865(119890119894) oplus 119865(120576

119894) = ℎ

119901

cup120574119894119901isin120583119894119901120574120576119894119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

120576119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2) 119865 (119890

119899) oplus 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574120576119894119901))

| 119901 = 1 2 119898

8 The Scientific World Journal

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1

times (119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897(119897minus1

times(

119899

sum

119894=1

120596119894119897 (120574120576119894119901)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + sum119899

119894=1120596119894119897 (120574120576119894119901))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

(21)

Thus the proof of Property 5 is completed

Definition 20 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWG

operator can be called hesitant fuzzy soft weighted geometricoperator and defined as

HFSWG (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

(22)

Theorem 21 Let 119865(119890119894) = ℎ

119895119901119895| 119895 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using HFSWG operator is a HFSN and

119867119865119878119882119866(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 (sum119899

119894=1120596119894119896 (120574119894119901))

| 119901 = 1 2 119898

(23)Similarly it is clear that theHFSWGoperator also satisfies

the properties that HFSWA operator has and the relativedetails are omitted here

Definition 22 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

sum119899

119894=1120596119894= 1 Then the GHFSWA operator can be called the

generalized hesitant fuzzy soft weighted averaging operatorand defined as

GHFSWA120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

(24)

In particular if 120582 = 1 then the GHFSWA operator isreduced to the HFSWA operator

Theorem 23 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWA operator is aHFSN and

119866119867119865119878119882119860120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119901))))))

| 119901 = 1 2 119898

(25)

Definition 24 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

The Scientific World Journal 9

sum119899

119894=1120596119894= 1 Then the GHFSWG operator can be called the

generalized hesitant fuzzy soft weighted geometric operatorand defined as

GHFSWG120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

(26)

In particular if 120582 = 1 then the GHFSWG operator isreduced to the HFSWG operator

Theorem 25 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWG operator is aHFSN and

119866119867119865119878119882119866120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119901))))))

|

119901 = 1 2 119898

(27)

Similarly it is clear that the GHFSWA and GHFSWGoperators possess the same properties that the HFSWAoperator has

Example 26 Suppose 119865(1198901) = ℎ

103 05 ℎ

204 06

and 119865(1198902) = ℎ

104 05 ℎ

208 09 are the elements of

the HFSS (119865 119864) and 120596 = (03 07) is the weight vector of119865(119890119894) (119894 = 1 2) which indicate the corresponding importance

degrees

Using Theorems 19 and 21 if we assign 119896(119905) = minus log(119905)then

HFSWA (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

1 minus prod2

119894=1(1 minus 120574

119894119901)120596119894

| 119901 = 1 2

= ℎ1

037 045 043 050

ℎ2

072 083 075 085

HFSWG (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

prod2

119894=1120574120596119894

119894119901

| 119901 = 1 2

= ℎ1

037 043 043 050

ℎ2

065 071 073 080

(28)

5 Multicriteria Group Decision MakingApproach with HFSSs

Consider a MCGDM with hesitant fuzzy soft informationand assume there are n alternatives119880 = ℎ

1 ℎ2 ℎ

119899 andm

criteria 119864 = 1198901 1198902 119890

119898 and the weight vector of criteria

is 120596 = (1205961 1205962 120596

119898) where 120596

119895ge 0 (119895 = 1 2 119898)

and sum119898

119895=1120596119895= 1 Suppose that there are 119896 decision-makers

119863 = 1198891 1198892 119889

119896 whose corresponding weight vector

is 120578 = (1205781 1205782 120578

119896) Then a MCGDM problem can be

concisely expressed by 119896HFSSs as follows

119865119896(119890119894) =

ℎ119895

120583119894119895

| 119895 = 1 2 119899 (119894 = 1 2 119898) (29)

To obtain the best alternative the steps of the proposedMCGDM approach with HFSSs are given as follows

Step 1 (normalize the decision information) For benefit-typecriteria no further action need be taken However for cost-type criteria the negation operator given in Definition 16need be used The normalized information can be denotedas follows

119865 (119890119894) =

ℎ119895

120583119894119895

119895 isin 119861119879

ℎ119895

cup120574119894119895isin120583119894119895

1 minus 120574119894119895

119895 isin 119862119879

(119894 = 1 2 119898 119895 = 1 2 119899)

(30)

Here 119861119879is the set of benefit-type criteria and119862

119879is the set

of cost-type criteria

Step 2 (aggregate the HFSEs of each decision-maker) Calcu-late the comprehensive evaluation values of each alternativefor each decision-maker by using the GHFSWAor GHFSWGoperator

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

6 The Scientific World Journal

=

ℎ119901

⋃1205741119901isin12058311199011205742119901isin1205832119901

119897minus1 (1205961119897 (1205741119901) + 1205962119897 (1205742119901))

|

119901 = 1 2 119898

(11)

Suppose (10) holds for 119899 = 119896 that is

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896))

=

119896

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119896) 119865 (119890

119896+1))

=

119896

⨁

119894=1

120596119894119865 (119890119894) oplus 120596119894+1

119865 (119890119894+1

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574119896+1119901isin120583119896+1119901

119897minus1 (120582119897 (120574119896+1119901

))

| 119901 = 1 2 119898

=

ℎ119901( ⋃

120574119894119901isin120583119894119901

119897minus1(119897(119897minus1(

119896

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897 (119897minus1(120596119896+1

119897 (120574119896+1119901

)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896

119894=1120596119894119897 (120574119894119901) + 120596119896+1

119897 (120574119896+1119901

))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119896+1

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

(12)

Now (10) holds for 119899 = 119896+1 and thus (10) holds for all 119899Subsequently some desirable properties of the HFSWA

operator are investigated

Property 1 If all 119865(119890119894) (119894 = 1 2 119899) are equal that is

119865(119890119894) = 119865(119890) = ℎ

119901120583119894119901| 119901 = 1 2 119898 for all 119894 then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = 119865 (119890) (13)

Proof Let 119865(119890119894) = 119865(119890) and then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= HFSWA (119865 (119890) 119865 (119890) 119865 (119890)) =

119899

⨁

119894=1

120596119894119865 (119890)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (119897 (120574119894119901))

| 119901 = 1 2 119898

=

ℎ119901

120583119894119901

| 119901 = 1 2 119898 (119894 = 1 2 119899) = 119865 (119890)

(14)

Property 2 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 119865(120576) =

ℎ119901120583120576119901

| 119901 = 1 2 119898 is a hesitant fuzzy soft element(HFSE) then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(15)

Proof Since 119865(119890119895) oplus 119865(120576) = ℎ

119901cup120574119895119901isin120583119895119901120574120576119901isin120583120576119901

119897minus1(119897(120574119895119901) +

119897(120574120576119901)) | 119901 = 1 2 119898 then

HFSWA (119865 (1198901) oplus 119865 (120576) 119865 (119890

2) oplus 119865 (120576) 119865 (119890

119899) oplus 119865 (120576))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119901)))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

120583120576119901

| 119901 = 1 2 119898

The Scientific World Journal 7

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119901isin120583120576119901

119897minus1(119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901))))

+ 119897 (120574120576119901))

minus1

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

(16)

Therefore HFSWA (119865(1198901) oplus 119865(120576) 119865(119890

2) oplus 119865(120576) 119865(119890

119899) oplus

119865(120576)) =HFSWA (119865(1198901) 119865(1198902) 119865(119890

119899)) oplus 119865(120576)

Property 3 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0

then

HFSWA (120582119865 (1198901) 120582119865 (119890

2) 120582119865 (119890

119899))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

(17)

Proof According to Definition 16 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1

(120582119897(120574119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (120582119897 (120574

119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894(120582119897 (120574

119894119901)))

| 119901 = 1 2 119898

120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582 (sum119899

119894=1120596119894119897 (120574119894119901)))

| 119901 = 1 2 119898

(18)

Therefore the proof of Property 3 is completed

According to Properties 2 and 3 Property 4 can be given

Property 4 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be all the elements of the HFSS (119865 119864) and let120596 = (120596

1 1205962 120596

119899) be the weight vector of 119865(119890

119894) such that

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0 and

119865(120576) = ℎ119901120583120576119901| 119901 = 1 2 119898 is a HFSE then

HFSWA (120582119865 (1198901) oplus 119865 (120576) 120582119865 (119890

2) oplus 119865 (120576)

120582119865 (119890119899) oplus 119865 (120576))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(19)

Property 5 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 1198641) let 119865(120576

119894) =

ℎ119901120583120576119894119901| 119901 = 1 2 119898 (119894 = 1 2 119899) be the elements of

HFSS (119865 1198642) and let120596 = (120596

1 1205962 120596

119899) be theweight vector

of them satisfying 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2)

119865 (119890119899) oplus 119865 (120576

119899))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

(20)

Proof According to Definition 16 119865(119890119894) oplus 119865(120576

119894) = ℎ

119901

cup120574119894119901isin120583119894119901120574120576119894119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

120576119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2) 119865 (119890

119899) oplus 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574120576119894119901))

| 119901 = 1 2 119898

8 The Scientific World Journal

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1

times (119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897(119897minus1

times(

119899

sum

119894=1

120596119894119897 (120574120576119894119901)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + sum119899

119894=1120596119894119897 (120574120576119894119901))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

(21)

Thus the proof of Property 5 is completed

Definition 20 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWG

operator can be called hesitant fuzzy soft weighted geometricoperator and defined as

HFSWG (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

(22)

Theorem 21 Let 119865(119890119894) = ℎ

119895119901119895| 119895 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using HFSWG operator is a HFSN and

119867119865119878119882119866(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 (sum119899

119894=1120596119894119896 (120574119894119901))

| 119901 = 1 2 119898

(23)Similarly it is clear that theHFSWGoperator also satisfies

the properties that HFSWA operator has and the relativedetails are omitted here

Definition 22 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

sum119899

119894=1120596119894= 1 Then the GHFSWA operator can be called the

generalized hesitant fuzzy soft weighted averaging operatorand defined as

GHFSWA120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

(24)

In particular if 120582 = 1 then the GHFSWA operator isreduced to the HFSWA operator

Theorem 23 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWA operator is aHFSN and

119866119867119865119878119882119860120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119901))))))

| 119901 = 1 2 119898

(25)

Definition 24 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

The Scientific World Journal 9

sum119899

119894=1120596119894= 1 Then the GHFSWG operator can be called the

generalized hesitant fuzzy soft weighted geometric operatorand defined as

GHFSWG120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

(26)

In particular if 120582 = 1 then the GHFSWG operator isreduced to the HFSWG operator

Theorem 25 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWG operator is aHFSN and

119866119867119865119878119882119866120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119901))))))

|

119901 = 1 2 119898

(27)

Similarly it is clear that the GHFSWA and GHFSWGoperators possess the same properties that the HFSWAoperator has

Example 26 Suppose 119865(1198901) = ℎ

103 05 ℎ

204 06

and 119865(1198902) = ℎ

104 05 ℎ

208 09 are the elements of

the HFSS (119865 119864) and 120596 = (03 07) is the weight vector of119865(119890119894) (119894 = 1 2) which indicate the corresponding importance

degrees

Using Theorems 19 and 21 if we assign 119896(119905) = minus log(119905)then

HFSWA (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

1 minus prod2

119894=1(1 minus 120574

119894119901)120596119894

| 119901 = 1 2

= ℎ1

037 045 043 050

ℎ2

072 083 075 085

HFSWG (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

prod2

119894=1120574120596119894

119894119901

| 119901 = 1 2

= ℎ1

037 043 043 050

ℎ2

065 071 073 080

(28)

5 Multicriteria Group Decision MakingApproach with HFSSs

Consider a MCGDM with hesitant fuzzy soft informationand assume there are n alternatives119880 = ℎ

1 ℎ2 ℎ

119899 andm

criteria 119864 = 1198901 1198902 119890

119898 and the weight vector of criteria

is 120596 = (1205961 1205962 120596

119898) where 120596

119895ge 0 (119895 = 1 2 119898)

and sum119898

119895=1120596119895= 1 Suppose that there are 119896 decision-makers

119863 = 1198891 1198892 119889

119896 whose corresponding weight vector

is 120578 = (1205781 1205782 120578

119896) Then a MCGDM problem can be

concisely expressed by 119896HFSSs as follows

119865119896(119890119894) =

ℎ119895

120583119894119895

| 119895 = 1 2 119899 (119894 = 1 2 119898) (29)

To obtain the best alternative the steps of the proposedMCGDM approach with HFSSs are given as follows

Step 1 (normalize the decision information) For benefit-typecriteria no further action need be taken However for cost-type criteria the negation operator given in Definition 16need be used The normalized information can be denotedas follows

119865 (119890119894) =

ℎ119895

120583119894119895

119895 isin 119861119879

ℎ119895

cup120574119894119895isin120583119894119895

1 minus 120574119894119895

119895 isin 119862119879

(119894 = 1 2 119898 119895 = 1 2 119899)

(30)

Here 119861119879is the set of benefit-type criteria and119862

119879is the set

of cost-type criteria

Step 2 (aggregate the HFSEs of each decision-maker) Calcu-late the comprehensive evaluation values of each alternativefor each decision-maker by using the GHFSWAor GHFSWGoperator

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

The Scientific World Journal 7

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119901isin120583120576119901

119897minus1(119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901))))

+ 119897 (120574120576119901))

minus1

| 119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119901isin120583120576119901

(119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + 119897 (120574

120576119901)))

|

119901 = 1 2 119898

(16)

Therefore HFSWA (119865(1198901) oplus 119865(120576) 119865(119890

2) oplus 119865(120576) 119865(119890

119899) oplus

119865(120576)) =HFSWA (119865(1198901) 119865(1198902) 119865(119890

119899)) oplus 119865(120576)

Property 3 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

such that 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0

then

HFSWA (120582119865 (1198901) 120582119865 (119890

2) 120582119865 (119890

119899))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

(17)

Proof According to Definition 16 120582119865(119890119894) = ℎ

119901cup120574119894119901isin120583119894119901

119897minus1

(120582119897(120574119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (120582119897 (120574

119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894(120582119897 (120574

119894119901)))

| 119901 = 1 2 119898

120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582119897 (119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (120582 (sum119899

119894=1120596119894119897 (120574119894119901)))

| 119901 = 1 2 119898

(18)

Therefore the proof of Property 3 is completed

According to Properties 2 and 3 Property 4 can be given

Property 4 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be all the elements of the HFSS (119865 119864) and let120596 = (120596

1 1205962 120596

119899) be the weight vector of 119865(119890

119894) such that

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 If 120582 gt 0 and

119865(120576) = ℎ119901120583120576119901| 119901 = 1 2 119898 is a HFSE then

HFSWA (120582119865 (1198901) oplus 119865 (120576) 120582119865 (119890

2) oplus 119865 (120576)

120582119865 (119890119899) oplus 119865 (120576))

= 120582HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) oplus 119865 (120576)

(19)

Property 5 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 1198641) let 119865(120576

119894) =

ℎ119901120583120576119894119901| 119901 = 1 2 119898 (119894 = 1 2 119899) be the elements of

HFSS (119865 1198642) and let120596 = (120596

1 1205962 120596

119899) be theweight vector

of them satisfying 120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2)

119865 (119890119899) oplus 119865 (120576

119899))

= HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

(20)

Proof According to Definition 16 119865(119890119894) oplus 119865(120576

119894) = ℎ

119901

cup120574119894119901isin120583119894119901120574120576119894119901isin120583119895119901

119897minus1(119897(120574119894119901) + 119897(120574

120576119894119901)) | 119901 = 1 2 119898

Then

HFSWA (119865 (1198901) oplus 119865 (120576

1) 119865 (119890

2) oplus 119865 (120576

2) 119865 (119890

119899) oplus 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (119897minus1 (119897 (120574

119894119901) + 119897 (120574

120576119894119901))))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

HFSWA (119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

oplusHFSWA (119865 (1205761) 119865 (120576

2) 119865 (120576

119899))

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901))

| 119901 = 1 2 119898

oplus

ℎ119901

⋃120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574120576119894119901))

| 119901 = 1 2 119898

8 The Scientific World Journal

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1

times (119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897(119897minus1

times(

119899

sum

119894=1

120596119894119897 (120574120576119894119901)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + sum119899

119894=1120596119894119897 (120574120576119894119901))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

(21)

Thus the proof of Property 5 is completed

Definition 20 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWG

operator can be called hesitant fuzzy soft weighted geometricoperator and defined as

HFSWG (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

(22)

Theorem 21 Let 119865(119890119894) = ℎ

119895119901119895| 119895 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using HFSWG operator is a HFSN and

119867119865119878119882119866(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 (sum119899

119894=1120596119894119896 (120574119894119901))

| 119901 = 1 2 119898

(23)Similarly it is clear that theHFSWGoperator also satisfies

the properties that HFSWA operator has and the relativedetails are omitted here

Definition 22 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

sum119899

119894=1120596119894= 1 Then the GHFSWA operator can be called the

generalized hesitant fuzzy soft weighted averaging operatorand defined as

GHFSWA120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

(24)

In particular if 120582 = 1 then the GHFSWA operator isreduced to the HFSWA operator

Theorem 23 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWA operator is aHFSN and

119866119867119865119878119882119860120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119901))))))

| 119901 = 1 2 119898

(25)

Definition 24 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

The Scientific World Journal 9

sum119899

119894=1120596119894= 1 Then the GHFSWG operator can be called the

generalized hesitant fuzzy soft weighted geometric operatorand defined as

GHFSWG120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

(26)

In particular if 120582 = 1 then the GHFSWG operator isreduced to the HFSWG operator

Theorem 25 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWG operator is aHFSN and

119866119867119865119878119882119866120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119901))))))

|

119901 = 1 2 119898

(27)

Similarly it is clear that the GHFSWA and GHFSWGoperators possess the same properties that the HFSWAoperator has

Example 26 Suppose 119865(1198901) = ℎ

103 05 ℎ

204 06

and 119865(1198902) = ℎ

104 05 ℎ

208 09 are the elements of

the HFSS (119865 119864) and 120596 = (03 07) is the weight vector of119865(119890119894) (119894 = 1 2) which indicate the corresponding importance

degrees

Using Theorems 19 and 21 if we assign 119896(119905) = minus log(119905)then

HFSWA (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

1 minus prod2

119894=1(1 minus 120574

119894119901)120596119894

| 119901 = 1 2

= ℎ1

037 045 043 050

ℎ2

072 083 075 085

HFSWG (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

prod2

119894=1120574120596119894

119894119901

| 119901 = 1 2

= ℎ1

037 043 043 050

ℎ2

065 071 073 080

(28)

5 Multicriteria Group Decision MakingApproach with HFSSs

Consider a MCGDM with hesitant fuzzy soft informationand assume there are n alternatives119880 = ℎ

1 ℎ2 ℎ

119899 andm

criteria 119864 = 1198901 1198902 119890

119898 and the weight vector of criteria

is 120596 = (1205961 1205962 120596

119898) where 120596

119895ge 0 (119895 = 1 2 119898)

and sum119898

119895=1120596119895= 1 Suppose that there are 119896 decision-makers

119863 = 1198891 1198892 119889

119896 whose corresponding weight vector

is 120578 = (1205781 1205782 120578

119896) Then a MCGDM problem can be

concisely expressed by 119896HFSSs as follows

119865119896(119890119894) =

ℎ119895

120583119894119895

| 119895 = 1 2 119899 (119894 = 1 2 119898) (29)

To obtain the best alternative the steps of the proposedMCGDM approach with HFSSs are given as follows

Step 1 (normalize the decision information) For benefit-typecriteria no further action need be taken However for cost-type criteria the negation operator given in Definition 16need be used The normalized information can be denotedas follows

119865 (119890119894) =

ℎ119895

120583119894119895

119895 isin 119861119879

ℎ119895

cup120574119894119895isin120583119894119895

1 minus 120574119894119895

119895 isin 119862119879

(119894 = 1 2 119898 119895 = 1 2 119899)

(30)

Here 119861119879is the set of benefit-type criteria and119862

119879is the set

of cost-type criteria

Step 2 (aggregate the HFSEs of each decision-maker) Calcu-late the comprehensive evaluation values of each alternativefor each decision-maker by using the GHFSWAor GHFSWGoperator

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

8 The Scientific World Journal

=

ℎ119901( ⋃

120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1

times (119897(119897minus1(

119899

sum

119894=1

120596119894119897 (120574119894119901)))

+ 119897(119897minus1

times(

119899

sum

119894=1

120596119894119897 (120574120576119894119901)))))

minus1

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894119897 (120574119894119901) + sum119899

119894=1120596119894119897 (120574120576119894119901))

|

119901 = 1 2 119898

=

ℎ119901

⋃120574119894119901isin120583119894119901120574120576119894119901isin120583120576119894119901

119897minus1 (sum119899

119894=1120596119894(119897 (120574119894119901) + 119897 (120574

120576119894119901)))

|

119901 = 1 2 119898

(21)

Thus the proof of Property 5 is completed

Definition 20 Let 119865(119890119894) = ℎ

119901120583119894119901

| 119901 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the HFSWG

operator can be called hesitant fuzzy soft weighted geometricoperator and defined as

HFSWG (119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

(22)

Theorem 21 Let 119865(119890119894) = ℎ

119895119901119895| 119895 = 1 2 119898 (119894 =

1 2 119899) be the elements of the HFSS (119865 119864) and let 120596 =

(1205961 1205962 120596

119899) be the weight vector of 119865(119890

119894) (119894 = 1 2 119899)

where 120596119894indicates the importance degree of 119865(119890

119894) satisfying

120596119894ge 0 (119894 = 1 2 119899) and sum

119899

119894=1120596119894= 1 Then the aggregated

value by using HFSWG operator is a HFSN and

119867119865119878119882119866(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) =

119899

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 (sum119899

119894=1120596119894119896 (120574119894119901))

| 119901 = 1 2 119898

(23)Similarly it is clear that theHFSWGoperator also satisfies

the properties that HFSWA operator has and the relativedetails are omitted here

Definition 22 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

sum119899

119894=1120596119894= 1 Then the GHFSWA operator can be called the

generalized hesitant fuzzy soft weighted averaging operatorand defined as

GHFSWA120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

= (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

(24)

In particular if 120582 = 1 then the GHFSWA operator isreduced to the HFSWA operator

Theorem 23 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWA operator is aHFSN and

119866119867119865119878119882119860120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899)) = (

119899

⨁

119894=1

120596119894(119865 (119890119894))120582

)

1120582

=

ℎ119901

⋃120574119894119901isin120583119894119901

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119901))))))

| 119901 = 1 2 119898

(25)

Definition 24 Let119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) and

The Scientific World Journal 9

sum119899

119894=1120596119894= 1 Then the GHFSWG operator can be called the

generalized hesitant fuzzy soft weighted geometric operatorand defined as

GHFSWG120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

(26)

In particular if 120582 = 1 then the GHFSWG operator isreduced to the HFSWG operator

Theorem 25 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWG operator is aHFSN and

119866119867119865119878119882119866120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119901))))))

|

119901 = 1 2 119898

(27)

Similarly it is clear that the GHFSWA and GHFSWGoperators possess the same properties that the HFSWAoperator has

Example 26 Suppose 119865(1198901) = ℎ

103 05 ℎ

204 06

and 119865(1198902) = ℎ

104 05 ℎ

208 09 are the elements of

the HFSS (119865 119864) and 120596 = (03 07) is the weight vector of119865(119890119894) (119894 = 1 2) which indicate the corresponding importance

degrees

Using Theorems 19 and 21 if we assign 119896(119905) = minus log(119905)then

HFSWA (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

1 minus prod2

119894=1(1 minus 120574

119894119901)120596119894

| 119901 = 1 2

= ℎ1

037 045 043 050

ℎ2

072 083 075 085

HFSWG (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

prod2

119894=1120574120596119894

119894119901

| 119901 = 1 2

= ℎ1

037 043 043 050

ℎ2

065 071 073 080

(28)

5 Multicriteria Group Decision MakingApproach with HFSSs

Consider a MCGDM with hesitant fuzzy soft informationand assume there are n alternatives119880 = ℎ

1 ℎ2 ℎ

119899 andm

criteria 119864 = 1198901 1198902 119890

119898 and the weight vector of criteria

is 120596 = (1205961 1205962 120596

119898) where 120596

119895ge 0 (119895 = 1 2 119898)

and sum119898

119895=1120596119895= 1 Suppose that there are 119896 decision-makers

119863 = 1198891 1198892 119889

119896 whose corresponding weight vector

is 120578 = (1205781 1205782 120578

119896) Then a MCGDM problem can be

concisely expressed by 119896HFSSs as follows

119865119896(119890119894) =

ℎ119895

120583119894119895

| 119895 = 1 2 119899 (119894 = 1 2 119898) (29)

To obtain the best alternative the steps of the proposedMCGDM approach with HFSSs are given as follows

Step 1 (normalize the decision information) For benefit-typecriteria no further action need be taken However for cost-type criteria the negation operator given in Definition 16need be used The normalized information can be denotedas follows

119865 (119890119894) =

ℎ119895

120583119894119895

119895 isin 119861119879

ℎ119895

cup120574119894119895isin120583119894119895

1 minus 120574119894119895

119895 isin 119862119879

(119894 = 1 2 119898 119895 = 1 2 119899)

(30)

Here 119861119879is the set of benefit-type criteria and119862

119879is the set

of cost-type criteria

Step 2 (aggregate the HFSEs of each decision-maker) Calcu-late the comprehensive evaluation values of each alternativefor each decision-maker by using the GHFSWAor GHFSWGoperator

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

The Scientific World Journal 9

sum119899

119894=1120596119894= 1 Then the GHFSWG operator can be called the

generalized hesitant fuzzy soft weighted geometric operatorand defined as

GHFSWG120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

(26)

In particular if 120582 = 1 then the GHFSWG operator isreduced to the HFSWG operator

Theorem 25 Let 119865(119890119894) (119894 = 1 2 119899) be the elements of the

HFSS (119865 119864) and let 120596 = (1205961 1205962 120596

119899) be the weight vector

of 119865(119890119894) (119894 = 1 2 119899) where 120596

119894indicates the importance

degree of 119865(119890119894) satisfying 120596

119894ge 0 (119894 = 1 2 119899) andsum119899

119894=1120596119894=

1 Then the aggregated value by using GHFSWG operator is aHFSN and

119866119867119865119878119882119866120582(119865 (1198901) 119865 (119890

2) 119865 (119890

119899))

=1

120582(

119899

⨁

119894=1

(120582119865 (119890119894))120596119894

)

=

ℎ119901

⋃120574119894119901isin120583119894119901

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119901))))))

|

119901 = 1 2 119898

(27)

Similarly it is clear that the GHFSWA and GHFSWGoperators possess the same properties that the HFSWAoperator has

Example 26 Suppose 119865(1198901) = ℎ

103 05 ℎ

204 06

and 119865(1198902) = ℎ

104 05 ℎ

208 09 are the elements of

the HFSS (119865 119864) and 120596 = (03 07) is the weight vector of119865(119890119894) (119894 = 1 2) which indicate the corresponding importance

degrees

Using Theorems 19 and 21 if we assign 119896(119905) = minus log(119905)then

HFSWA (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

120596119894119865 (119890119894)

=

ℎ119901

⋃120574119894119901isin120583119894119901

1 minus prod2

119894=1(1 minus 120574

119894119901)120596119894

| 119901 = 1 2

= ℎ1

037 045 043 050

ℎ2

072 083 075 085

HFSWG (119865 (1198901) 119865 (119890

2)) =

2

⨁

119894=1

119865 (119890119894)120596119894

=

ℎ119901

⋃120574119894119901isin120583119894119901

prod2

119894=1120574120596119894

119894119901

| 119901 = 1 2

= ℎ1

037 043 043 050

ℎ2

065 071 073 080

(28)

5 Multicriteria Group Decision MakingApproach with HFSSs

Consider a MCGDM with hesitant fuzzy soft informationand assume there are n alternatives119880 = ℎ

1 ℎ2 ℎ

119899 andm

criteria 119864 = 1198901 1198902 119890

119898 and the weight vector of criteria

is 120596 = (1205961 1205962 120596

119898) where 120596

119895ge 0 (119895 = 1 2 119898)

and sum119898

119895=1120596119895= 1 Suppose that there are 119896 decision-makers

119863 = 1198891 1198892 119889

119896 whose corresponding weight vector

is 120578 = (1205781 1205782 120578

119896) Then a MCGDM problem can be

concisely expressed by 119896HFSSs as follows

119865119896(119890119894) =

ℎ119895

120583119894119895

| 119895 = 1 2 119899 (119894 = 1 2 119898) (29)

To obtain the best alternative the steps of the proposedMCGDM approach with HFSSs are given as follows

Step 1 (normalize the decision information) For benefit-typecriteria no further action need be taken However for cost-type criteria the negation operator given in Definition 16need be used The normalized information can be denotedas follows

119865 (119890119894) =

ℎ119895

120583119894119895

119895 isin 119861119879

ℎ119895

cup120574119894119895isin120583119894119895

1 minus 120574119894119895

119895 isin 119862119879

(119894 = 1 2 119898 119895 = 1 2 119899)

(30)

Here 119861119879is the set of benefit-type criteria and119862

119879is the set

of cost-type criteria

Step 2 (aggregate the HFSEs of each decision-maker) Calcu-late the comprehensive evaluation values of each alternativefor each decision-maker by using the GHFSWAor GHFSWGoperator

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

10 The Scientific World Journal

119865119896(119890) = GHFSWA

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) = (

119899

⨁

119894=1

120596119894(119865119896(119890119894))120582

)

1120582

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 ((1120582) 119896 (119897minus1 (sum119899

119894=1120596119894119897 (119896minus1 (120582119896 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(31)

or

119865119896(1198901015840) = GHFSWG

120582(119865119896(1198901) 119865119896(1198902) 119865

119896(119890119899)) =

1

120582(

119899

⨁

119894=1

(120582119865119896(119890119894))120596119894

)

=

ℎ119895

⋃120574119894119895isin120583119894119895

119897minus1 ((1120582) 119897 (119896minus1 (sum119899

119894=1120596119894119896 (119897minus1 (120582119897 (120574

119894119895))))))

| 119895 = 1 2 119898

(119894 = 1 2 119899)

(32)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) Calculate the overall values by using the HFSWAor HFSWG operator

119865 (119890) = HFSWA (1198651(119890) 1198652(119890) 119865

119896(119890)) =

119896

⨁

119894=1

120578119894119865119894(119890)

=

ℎj

⋃120574119894119895isin120583119894119895

119897minus1 (sum119896

119894=1120578119894119897 (120574119894119895))

| 119895 = 1 2 119898

(33)

or

119865 (119890) = HFSWG (1198651(1198901015840) 1198652(1198901015840) 119865

119896(1198901015840))

=

119896

⨁

119894=1

119865119896(1198901015840)120578119894

=

ℎ119895

⋃120574119894119895isin120583119894119895

119896minus1 (sum119896

119894=1120578119894119896 (120574119894119895))

| 119895 = 1 2 119898

(34)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 119899) of the alternative

ℎ119894(119894 = 1 2 119899) by using the ranking method described in

Definitions 11ndash13

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 120583

119894(119894 = 1 2 119899)

6 Illustrative Example

In this section we cite the commonly used example [27] andextend it to the hesitant fuzzy soft environment

Mr Xrsquos family wants to buy an attractive house consid-ering three criteria being beautiful (119890

1) being cheap (119890

2) and

being in the green surroundings (1198903) respectivelyTheweight

of the criteria is 120596 = (03 03 04) The family can make theirchoice among four houses 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 Mr X his

wife and his son have their own opinions about the givenhouses whose corresponding weight vector is (05 03 02)They make their evaluations for four houses according tothree criteria 119864 = 119890

1 1198902 1198903 and the evaluation values are in

the form of HFSSs as shown in Tables 4 5 and 6

61 The Decision Making Procedure Based on HFSSs Toobtain the best alternative let 120582 = 1 119896(119905) = minus log(119905) andthen the following steps are given

Step 1 (normalize the decision information) The criteria areof benefit-type so no transformation is required

Step 2 (aggregate the HFSEs of each decision-maker) Byusing the GHFSWA operator the comprehensive evaluationvalues of Mr X are

1198651(119890)

= GHFSWA (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0568 0468 0551 0448

ℎ2

0519 0493 0457 0427

ℎ3

0741 0597 0717 0561 0731 0574 0707 0545

ℎ4

0645

(35)

The comprehensive evaluation values of Mrs X are

1198652(119890) = GHFSWA (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0848

ℎ2

0703 0650

ℎ3

0482

ℎ4

0735

(36)

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

The Scientific World Journal 11

Table 4 Mr Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

03 02 09 08 01

ℎ2

08 07 03 03 02

ℎ3

02 01 07 06 09 07

ℎ4

07 01 08

Table 5 Mrs Xrsquos evaluation for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

08 08 09

ℎ2

05 07 08 07

ℎ3

07 04 03

ℎ4

07 09 05

Table 6 The evaluation of Mr X for the alternative houses

119880 Beautiful Cheap In the green surroundingsℎ1

07 05 08 07

ℎ2

08 08 05

ℎ3

07 01 06

ℎ4

05 04 09 07

The comprehensive evaluation values of Mr Xrsquos son are

1198653(119890) = GHFSWA (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0703 0650

ℎ2

0711

ℎ3

0532

ℎ4

0748 0734

(37)

By using the GHFSWG operator the comprehensiveevaluation values of Mr X are

1198651(1198901015840)

= GHFSWG (1198651(1198901) 1198651(1198902) 1198651(1198903))

= ℎ1

0269 0259 0238 0230

ℎ2

0403 0342 0387 0329

ℎ3

0532 0481 0508 0459 0432 0390 0412 0373

ℎ4

0412

(38)

The comprehensive evaluation values of Mrs X are

1198652(1198901015840) = GHFSWG (119865

2(1198901) 1198652(1198902) 1198652(1198903))

= ℎ1

0839

ℎ2

0668 0633

ℎ3

0422

ℎ4

0660

(39)

The comprehensive evaluation values of Mr Xrsquos son are

1198653(1198901015840) = GHFSWG (119865

3(1198901) 1198653(1198902) 1198653(1198903))

= ℎ1

0668 0633

ℎ2

0663

ℎ3

0367

ℎ4

0682 0638

(40)

Step 3 (aggregate the comprehensive HFSEs of all decisionndashmakers) By utilizing the HFSWA operator the overall HFSS119865(119890) can be obtained as follows

119865 (119890)

= HFSWA (1198651(119890) 1198652(119890) 1198653(119890))

= ℎ1

0707 0697 0675 0664 0701 0691 0669 0658

ℎ2

0624 0605 0614 0595 0600 0580 0590 0569

ℎ3

0641 0552 0625 0533 0634 0540 0618 0524

ℎ4

0696 0693

(41)

By utilizing HFSWG operator the overall HFSS 119865(1198901015840) canbe obtained as follows

119865 (1198901015840)

= HFSWG (1198651(1198901015840) 1198652(1198901015840) 1198653(1198901015840))

= ℎ1

0454 0449 0445 0441 0427 0422 0420 0415

ℎ2

0518 0510 0477 0470 0508 0500 0468 0461

ℎ3

0461 0438 0450 0428 0415 0395 0405 0386

ℎ4

0525 0518

(42)

Step 4 Rank the HFNs 120583119894(119894 = 1 2 4) of the alternative

ℎ119894(119894 = 1 2 4) by using the ranking method described in

Definitions 11ndash13

The ranking results of 119904(120583119894) (119894 = 1 2 3 4) are shown in

Table 7

Step 5 Rank all the alternatives and select the best one(s) inaccordance with the ranking of 119904(120583

119894) (119894 = 1 2 4)

Obviously the best alternative is ℎ4 and the worst

alternative is ℎ3

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

12 The Scientific World Journal

Table 7 Ranking of the alternatives by utilizing 119896(119905) = minus log(119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 8 Ranking of the alternatives by utilizing 119896(119905) = log((2minus119905)119905)

119904(1205831) 119904(120583

2) 119904(120583

3) 119904(120583

4) Ranking

HFSWA 0658 0582 0563 0680 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

HFSWG 0474 0506 0447 0555 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

Table 9 Ranking of the alternatives by utilizing different 120582 inGHFSWA

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0683 0597 0583 0695 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 2 0710 0617 0612 0716 119904(ℎ4) gt 119904(ℎ

1) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 5 0756 0664 0662 0752 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

120582 = 10 0788 0703 0698 0783 119904(ℎ1) gt 119904(ℎ

4) gt 119904(ℎ

2) gt 119904(ℎ

3)

Table 10 Ranking of the alternatives by utilizing different 120582 inGHFSWG

120582 119904(ℎ1) 119904(ℎ

2) 119904(ℎ

3) 119904(ℎ

4) The final ranking

120582 = 1 0434 0489 0422 0522 119904(ℎ4) gt 119904(ℎ

2) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 2 0409 0470 0385 0462 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 5 0366 0437 0309 0378 119904(ℎ2) gt 119904(ℎ

4) gt 119904(ℎ

1) gt 119904(ℎ

3)

120582 = 10 0334 0399 0255 0318 119904(ℎ2) gt 119904(ℎ

1) gt 119904(ℎ

4) gt 119904(ℎ

3)

62 Sensitivity Analysis and Discussion In this subsectiondifferent t-norms as well as the parameter 120582 are adopted tocalculate the same example given in Section 61

Case 1 If Einstein t-norm and t-conorm are used to handlehesitant fuzzy soft information then 119896(119905) = log((2 minus 119905)119905) andthe ranking results of 119904(ℎ

119894) (119894 = 1 2 3 4) are shown in Table 8

Obviously the rankings of the alternatives in Table 8 gotby using 119896(119905) = log((2 minus 119905)119905) are completely the same asthose got by using 119896(119905) = minus log(119905) shown in Table 7 So usingdifferent t-norms could lead to no influence on the results

Case 2 In the following in order to illustrate the influence ofthe parameter 120582 on this example we use different values of 120582in Step 2 of the proposed approach to rank the alternativesAnd the ranking results are shown in Tables 9 and 10

From Tables 9 and 10 it can be seen that if the valueof 120582 is one then the best alternative is ℎ

4and the worst

alternative is ℎ3 no matter which operators are used in Step

2 of the proposed approach However either the GHFSWAor GHFSWA operator is used in Step 2 and different 120582 forthe same operator may lead to different aggregation results

and the final ranking of alternatives is also different as theparameter changes On the other hand if 120582 remains thesame the results obtained by using the GHFSWA operatorare also different from those by the GHFSWG operator Byusing theGHFSWAoperator decision-makers emphasize thecomprehensive priority of all criteria By using the GHFSWGoperator decision-makers avoid choosing the alternativesthat have poor performances in some criteria Apparentlyregardless of using different operators or different120582 theworstalternative is always ℎ

3while the best alternative may be

one of the other three alternatives Thus Mr Xrsquos family canproperly select the desirable alternative according to theirinterests and the actual needs

In a word the developed approach based on differentt-norms and t-conorms can be used to deal with differentrelationships among the aggregated arguments It can handleMCGDM problems in a flexible and objective manner underhesitant fuzzy soft environment and canprovidemore choicesfor decision-makers At the same time different resultsmay be obtained by using different aggregation operatorswhich reflected the preferences of decision-makers Themain advantages of the approach proposed in this paper arenot only its ability to effectively deal with the preferenceinformation expressed by HFSNs but also its considerationthat different t-norm and t-conorms can lead to differentaggregation operators This can avoid losing and distortingthe preference information provided which makes the finalresults suitably correspond with real life decision makingproblems

7 Conclusion

In this paper we introduced hesitant fuzzy soft sets whichcan be regarded as an extension of both a SS and a HFSHFSSs can describe the real preferences of decision-makersand reflect their uncertainty and hesitancy which permit themembership degree to have a set of possible values in SSsThis paper also focuses on MCGDM problems in which thepreferences of decision-makers are expressed by HFSNs Thenewapproach to solving these problems is based on the aggre-gation operators of HFSNs Having reviewed the relevant lit-eratures operations ofHFSNs based onArchimedean t-normand Archimedean t-conorm are defined Then four aggrega-tion operators such as the HFSWA HFSWG GHFSWA andGHFSWA operators are developed Different t-norms anddifferent parameters in generalized operators can be used inMCGDMmethods Finally a numerical example is providedto illustrate the advantages of the proposed approach Fromthe example it can be seen that HFSSs could vividly andeffectively represent various preferences of different decision-makers and avoid overlooking any subjective intentions ofdecision-makers In the future work HFSSs can be used todeal with much more uncertain problems

Conflict of Interests

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

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

The Scientific World Journal 13

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by the National Natural Science Foundation ofChina (nos 71271218 and 71221061)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[3] R R Yager ldquoMultiple objective decision-making using fuzzysetsrdquo International Journal of Man-Machine Studies vol 9 no4 pp 375ndash382 1977

[4] L A Zadeh ldquoFuzzy logic and approximate reasoningrdquo Synthesevol 30 no 3-4 pp 407ndash428 1975

[5] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2 pp 121ndash1461990

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

[7] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

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

[9] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[10] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[11] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[12] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[13] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on Atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[14] J-Q Wang Z-Q Han and H-Y Zhang ldquoMulti-criteria groupdecision-making method based on intuitionistic interval fuzzyinformationrdquoGroup Decision and Negotiation vol 23 no 4 pp715ndash733 2014

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

[16] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

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

[18] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012

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

[20] 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

[21] J-Q Wang D-D Wang H Y Zhang and X H Chen ldquoMulti-criteria outranking approach with hestitant fuzzy setsrdquo ORSpectrum vol 36 no 4 pp 1001ndash1019 2013

[22] D Yu Y Wu and W Zhou ldquoMulti-criteria decision makingbased on Choquet integral under hesitant fuzzy environmentrdquoJournal of Computational Information Systems vol 7 no 12 pp4506ndash4513 2011

[23] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013

[24] Z S Xu and M M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

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

[26] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[27] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[28] PKMaji A R Roy andR Biswas ldquoAn application of soft sets ina decision making problemrdquo Computers and Mathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[29] M I Ali F Feng X Liu W Min andM Shabir ldquoOn some newoperations in soft set theoryrdquo Computers and Mathematics withApplications vol 57 no 9 pp 1547ndash1553 2009

[30] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

[31] Y C Jiang Y Tang Q M Chen and Z Cao ldquoSemanticoperations of multiple soft sets under conflictrdquo Computers ampMathematics with Applications vol 62 no 4 pp 1923ndash19392011

[32] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[33] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[34] U Acar F Koyuncu and B Tanay ldquoSoft sets and soft ringsrdquoComputers amp Mathematics with Applications vol 59 no 11 pp3458ndash3463 2010

[35] K Gong Z Xiao and X Zhang ldquoThe bijective soft set with itsoperationsrdquo Computers andMathematics with Applications vol60 no 8 pp 2270ndash2278 2010

[36] K V Babitha and J J Sunil ldquoSoft set relations and functionsrdquoComputers amp Mathematics with Applications vol 60 no 7 pp1840ndash1849 2010

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013

14 The Scientific World Journal

[37] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[38] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing Journal vol 12 no 6 pp 1814ndash18212012

[39] N Cagman and S Enginoglu ldquoSoft set theory and uni-intdecision makingrdquo European Journal of Operational Researchvol 207 no 2 pp 848ndash855 2010

[40] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[41] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[42] M Borah T J Neog andD K Sut ldquoA study on some operationsof fuzzy soft setsrdquo International Journal of Modern EngineeringResearch vol 2 no 2 pp 219ndash225 2012

[43] X C Guan Y Li and F Feng ldquoA new order relation on fuzzysoft sets and its applicationrdquo Soft Computing vol 17 no 1 pp63ndash70 2013

[44] F Feng H Fujita Y B Jun and M Khan ldquoDecompositionof fuzzy soft sets with finite value spacesrdquo The Scientific WorldJournal vol 2014 Article ID 902687 10 pages 2014

[45] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[46] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[47] X Zhou and Q Li ldquoGeneralized vague soft set and its latticestructuresrdquo Journal of Computational Analysis and Applicationsvol 17 no 2 pp 265ndash271 2014

[48] X B Yang T Y Lin J Yang Y Li and D Yu ldquoCombinationof interval-valued fuzzy set and soft setrdquo Computers andMathematics with Applications vol 58 no 3 pp 521ndash527 2009

[49] X Ma H Qin N Sulaiman T Herawan and J H AbawajyldquoThe parameter reduction of the interval-valued fuzzy soft setsand its related algorithmsrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 1 pp 57ndash71 2014

[50] Y Jiang Y Tang H Liu and Z Chen ldquoEntropy on intuitionisticfuzzy soft sets and on interval-valued fuzzy soft setsrdquo Informa-tion Sciences vol 240 pp 95ndash114 2013

[51] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[52] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[53] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[54] Z Kong L Wang and Z Wu ldquoApplication of fuzzy soft setin decision making problems based on grey theoryrdquo Journal ofComputational and Applied Mathematics vol 236 no 6 pp1521ndash1530 2011

[55] T Mitra Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquoApplied Soft Computing Journal vol 12 no 10pp 3260ndash3275 2012

[56] Z Xiao W Chen and L Li ldquoAn integrated FCM and fuzzysoft set for supplier selection problem based on risk evaluationrdquo

Applied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 36 no 4 pp1444ndash1454 2012

[57] Z W Li G Q Wen and Y Han ldquoDecision making basedon intuitionistic fuzzy soft sets and its algorithmrdquo Journal ofComputational Analysis and Applications vol 17 no 4 pp 620ndash631 2014

[58] N Cagman and S Karatas ldquoIntuitionistic fuzzy soft set theoryand its decision makingrdquo Journal of Intelligent amp Fuzzy Systemsvol 24 no 4 pp 829ndash836 2013

[59] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[60] Z Xiao X Yang Q Niu et al ldquoA new evaluation method basedon D-S generalized fuzzy soft sets and its application inmedicaldiagnosis problemrdquo Applied Mathematical Modelling vol 36no 10 pp 4592ndash4604 2012

[61] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling vol 36 no 12 pp 5844ndash5855 2012

[62] H Zhang L Shu and S Liao ldquoGeneralized trapezoidal fuzzysoft set and its application in medical diagnosisrdquo Journal ofApplied Mathematics vol 2014 Article ID 312069 12 pages2014

[63] Z Zhang ldquoA rough set approach to intuitionistic fuzzy soft setbased decision makingrdquo Applied Mathematical Modelling vol36 no 10 pp 4605ndash4633 2012

[64] K V Babitha and S J John ldquoHesitant fuzzy soft setsrdquo Journal ofNew Results in Science vol 3 pp 98ndash107 2013

[65] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[66] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[67] E P Klement and R Mesiar Eds Logical Algebraic Analyticand Probabilistic Aspects of Triangular Norms Elsevier NewYork NY USA 2005

[68] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Heidelberg Germany 2007

[69] NChen Z Xu andMXia ldquoThe electre Imulti-critria decision-making method based on hesitant fuzzy setsrdquo InternationalJournal of Information Technology and Decision Making vol 13no 2 pp 1ndash37 2013