research article linguistic intuitionistic fuzzy sets and...
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Research ArticleLinguistic Intuitionistic Fuzzy Sets and Application in MAGDM
Huimin Zhang
School of Management Henan University of Technology Zhengzhou 450001 China
Correspondence should be addressed to Huimin Zhang zhm76126com
Received 19 January 2014 Revised 5 April 2014 Accepted 8 April 2014 Published 29 April 2014
Academic Editor Reinaldo Martinez Palhares
Copyright copy 2014 Huimin Zhang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
To better deal with imprecise and uncertain information in decision making the definition of linguistic intuitionistic fuzzysets (LIFSs) is introduced which is characterized by a linguistic membership degree and a linguistic nonmembership degreerespectively To compare any two linguistic intuitionistic fuzzy values (LIFVs) the score function and accuracy function aredefined Then based on 119905-norm and 119905-conorm several aggregation operators are proposed to aggregate linguistic intuitionisticfuzzy information which avoid the limitations in exiting linguistic operation In addition the desired properties of these linguisticintuitionistic fuzzy aggregation operators are discussed Finally a numerical example is provided to illustrate the efficiency of theproposed method in multiple attribute group decision making (MAGDM)
1 Introduction
Intuitionistic fuzzy set (IFS) [1] which is characterized bya degree of membership and a degree of nonmembershipis a very powerful tool to process vague information Afterthe pioneering study of Atanassov [1] the IFS has capturedmuch attention from researchers in various fields and manyachievements have been made such as entropy measure ofIFS [2ndash7] distance or similarity measure between IFSs [8ndash13] and aggregation operators of IFS [14ndash21] In additionrelated to IFS some authors proposed several other toolsto handle vague and imprecise information whereby twoor more sources of vagueness appear simultaneously [22]Atanassov andGargov [23] introduced the notion of interval-valued intuitionistic fuzzy set (IVIFS) which is characterizedby a membership function and a nonmembership functionwith interval values Torra [24] and Torra and Narukawa [25]gave a definition of hesitant fuzzy set (HFS) which can betterdeal with the situations where several values are possible todetermine the membership of an element Zhu et al [26]defined dual hesitant fuzzy set in terms of two functions thatreturn two sets of membership values and nonmembershipvalues respectively for each element in the domain
Although the foregoing fuzzy tools are suitable fordealing with problems that are defined as quantitative sit-uations [22] uncertainty is often because of the vaguenessof meanings that is used by experts in problems whose
nature is rather qualitative For example for reason of theincreasing complexity of the decision making environmenttime pressure and the lack of data or knowledge aboutthe problem domain in the process of decision makingunder intuitionistic fuzzy environment a decision makermay have difficulty in expressing the degree of membershipand nonmembership as exact values whereas he or she maythink the use of linguistic values is more straightforwardand suitable to express the degree of membership and non-membership Similar to IFS linguistic intuitionistic fuzzy set(LIFS) is characterized by a linguisticmembership degree anda linguistic nonmembership degree respectively By usingthe LIFS decision makers are able to consider a linguistichesitancy degree in the belongingness of an element to a setwhere they cannot easily express their subjective judgmentwith a single linguistic term
The outline of the paper is organized as follows Thefollowing section presents a brief introduction to the basicknowledge that will be used in the definition of LIFSSection 3 gives the concept of LIFS and constructs thescore function and accuracy function for LIFS Section 4develops several aggregation operators for LIFS Section 5proposes a MAGDM method with linguistic intuitionisticfuzzy information In Section 6 an application of the newapproach is presented Finally conclusions are provided inSection 7
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 432092 11 pageshttpdxdoiorg1011552014432092
2 Journal of Applied Mathematics
2 Preliminaries
In the following some basic concepts and knowledge relatedto IFS and linguistic approach are briefly described
Definition 1 (see [1]) Let119883 be a universal set An IFS 119860 in119883
is given as
119860 = (119909 120583119860 (119909) V119860 (119909)) | 119909 isin 119883 (1)
where the functions 120583119860(119909) 119883 rarr [0 1] V
119860(119909) 119883 rarr [0 1]
stand for the degree of membership and nonmembershipof the element 119909 to 119860 respectively Any 119909 isin 119883 meets thecondition 0 le 120583
119860(119909) + V
119860(119909) le 1
120587119860(119909) is called intuitionistic index or degree of indeter-
minacy of 119909 to 119860 120587119860(119909) = 1 minus 120583
119860(119909) minus V
119860(119909) Obviously if
120587119860(119909) = 0 IFS 119860 is reduced to a fuzzy setSome basic definitions and operations on IFS are pre-
sented as follows
Definition 2 (see [14 15]) If 119860 and 119861 are two IFSs of the set119883 then
(1) 119860 = 119861 if and only if forall119909 isin 119883 120583119860(119909) = 120583
119861(119909) and
V119860(119909) = V
119861(119909)
(2) 119860119888 = (119909 V119860(119909) 120583119860(119909)) | 119909 isin 119883 where 119860
119888 is thecomplement of 119860
(3) 119860 and 119861 = (min(120583119860(119909) 120583119861(119909))max(V
119860(119909) V119861(119909)))
(4) 119860 or 119861 = (max(120583119860(119909) 120583119861(119909))min(V
119860(119909) V119861(119909)))
(5) 119860 oplus 119861 = (120583119860(119909) + 120583
119861(119909) minus 120583
119860(119909)120583119861(119909) V119860(119909)V119861(119909))
(6) 119860 otimes 119861 = (120583119860(119909)120583119861(119909) V119860(119909) + V
119861(119909) minus V
119860(119909)V119861(119909))
(7) 120582119860 = (1 minus (1 minus 120583119860(119909))120582 V119860(119909)120582) 120582 gt 0
(8) 119860120582 = (120583119860(119909)120582 1 minus (1 minus V
119860(119909))120582) 120582 gt 0
In real world many decision making problems presentqualitative aspects that are complex to assess by means ofnumerical values In such cases it may be more suitable toconsider them as linguistic variables
Let 119878 = 119904119894| 119894 = 0 1 119905 be a finite linguistic term set
with odd cardinality where 119904119894represents a possible linguistic
term for a linguistic variable For example a set of seven terms119878 can be expressed as follows
119878 = 1199040= N (none) 1199041 = VL (very low) 119904
2= L (low)
1199043= M (medium) 1199044 = H (high)
1199045= VH (very high) 119904
6= P (perfect)
(2)
It is required that the linguistic term set should satisfy thefollowing characteristics [27ndash30]
(1) The set is ordered 119904119894gt 119904119895 if and only if 119894 gt 119895
(2) There is a negation operator Neg(119904119894) = 119904119895such that
119895 = 119905 minus 119894(3) Max operator max(119904
119894 119904119895) = 119904119894 if and only if 119894 ge 119895
(4) Min operator min(119904119894 119904119895) = 119904119894 if and only if 119894 le 119895
To preserve all the given information Xu [31] extendedthe discrete term set 119878 to a continuous linguistic term set 119878 =
119904120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] where if 119904
120572isin 119878 then 119904
120572is called
the original linguistic term Otherwise 119904120572is called the virtual
linguistic term
Definition 3 (see [31 32]) Consider any two linguistic terms119904120572 119904120573
isin 119878 and 120583 1205831 1205832
isin [0 1] the add and multiplyoperations of linguistic variable are defined as follows
119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904120572+120573
119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904120572120573
(3)
119905-norm and 119905-conorm have been widely used to constructoperations for fuzzy sets and IFSs
Definition 4 (see [33 34]) A 119905-norm is a mapping 119879 [0 1] times
[0 1] rarr [0 1] satisfying for all 119909 119910 119911 isin [0 1]
(1) 119879(119909 1) = 119909
(2) 119879(119909 119910) = 119879(119910 119909)
(3) 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)
(4) 119879(119909 119910) le 119879(119909 119911) whenever 119910 le 119911
The four basic 119905-norms 119879119872 119879119875 119879119871 and 119879
119863are given as
follows
119879119872(119909 119910) = min(119909 119910) (lattice operation)
119879119875(119909 119910) = 119909 sdot 119910 (algebraic operation)
119879119871(119909 119910) = max(119909+119910minus1 0) (Lukasiewicz operation)
119879119863(119909 119910) =
0 if (119909119910)isin[01)2min(119909119910) otherwise (drastic operation)
Definition 5 (see [33 34]) A t-conorm is a mapping 119878
[0 1] times [0 1] rarr [0 1] satisfying for all 119909 119910 119911 isin [0 1]
(1) 119878(119909 0) = 119909
(2) 119878(119909 119910) = 119878(119910 119909)
(3) 119878(119909 119878(119910 119911)) = 119878(119878(119909 119910) 119911)
(4) 119878(119909 119910) le 119878(119909 119911) whenever 119910 le 119911
The four basic 119905-conorms 119878119872 119878119875 119878119871 and 119878
119863are given as
follows
119878119872(119909 119910) = max(119909 119910) (lattice operation)
119878119875(119909 119910) = 119909 + 119910 minus 119909 sdot 119910 (algebraic operation)
119878119871(119909 119910) = min(119909 + 119910 1) (Lukasiewicz operation)
119878119863(119909 119910) =
max(119909119910) if (119909119910)isin(01]21 otherwise
(drastic operation)
3 Linguistic Intuitionistic Fuzzy Set
The concept of linguistic intuitionistic fuzzy set (LIFS) isgiven as follows
Journal of Applied Mathematics 3
Definition 6 Let119883 be a finite universal set and 119878 = 119904120572| 1199040lt
119904120572le 119904119905 120572 isin [0 119905] a continuous linguistic term set A LIFS 119860
in119883 is given as
119860 = (119909 119904120579 (119909) 119904120590 (119909)) | 119909 isin 119883 (4)
where 119904120579(119909) 119904120590(119909) isin 119878 stand for the linguistic membership
degree and linguistic nonmembership of the element 119909 to 119860respectively
For any 119909 isin 119883 the condition 0 le 120579 + 120590 le 119905 is alwayssatisfied 120587(119909) is called linguistic indeterminacy degree of 119909to 119860 120587(119909) = 119904
119905minus120579minus120590 Obviously if 120579 + 120590 = 119905 then LIFS
119860 has the minimum linguistic indeterminacy degree thatis 120587(119909) = 119904
0 which means the membership degree of 119909 to
119860 can be precisely expressed with a single linguistic termand LIFS 119860 is reduced to a linguistic variable Oppositelyif 120579 = 120590 = 0 then LIFS 119860(119909) has the maximum linguisticindeterminacy degree that is 120587(119909) = 119904
119905 Similar to IFS
the LIFS 119860(119909) can be transformed into an interval linguisticvariable [119904
120579(119909) 119904119905minus120590
(119909)] which indicates that the minimumandmaximum linguisticmembership degrees of the elements119909 to 119860 are 119904
120579and 119904119905minus120590
respectivelyFor notational simplicity we suppose both LIFS 119860 and
119861 contain only one element which stand for linguisticintuitionistic fuzzy values (LIFVs) that is the pairs 119860 =
(119904120579 119904120590) and 119861 = (119904
120583 119904])
To compare any two LIFVs the score function andaccuracy function are defined as follows
Definition 7 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
with 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] The score
function of 119860 is defined as
119878 (119860) = 119904(119905+120579minus120590)2
(5)
and the accuracy function is defined as
119867(119860) = 119904120579+120590
(6)
Thus 119860 and 119861 can be ranked by the following procedure
(1) if 119878(119860) gt 119878(119861) then 119860 gt 119861(2) if 119878(119860) = 119878(119861) and
(a) 119867(119860) = 119867(119861) then 119860 = 119861(b) 119867(119860) gt 119867(119861) then 119860 gt 119861(c) 119867(119860) lt 119867(119861) then 119860 lt 119861
It is easy to see that 0 le (119905+120579minus120590)2 le 119905 and 0 le 120579+120590 le 119905which means 119904
(119905+120579minus120590)2 119904120579+120590
isin 119878
Example 8 Let 119860 = (1199041 1199043) 119861 = (119904
0 1199044) and 119862 = (119904
1 1199045) be
LIFVs which are derived from 119878 = 119904120572| 1199040lt 119904120572le 1199046 120572 isin
[0 6]Applying formulas (5) and (6) we have
119878 (119860) = 1199042gt 119878 (119861) = 119878 (119862) = 119904
1
119867 (119861) = 1199044lt 119867 (119862) = 119904
6
(7)
Thus we obtain 119860 gt 119862 gt 119861
4 Aggregation Operators for LinguisticIntuitionistic Fuzzy Sets
Since the definition of LIFS is given it is necessary tointroduce the operations and computations between them
Definition 9 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
then
(1) 119860 = 119861 if and only if 119904120579= 119904120583and 119904120590= 119904]
(2) 119860119888 = (119904120590 119904120579) where 119860119888 is the complement of 119860
(3) the intersection of 119860 and 119861 119860 and 119861 =
(min(119904120579 119904120583)max(119904
120590 119904]))
(4) the union of 119860 and 119861 119860 or 119861 =
(max(119904120579 119904120583)min(119904
120590 119904]))
Motivated by 119905-norm and 119905-conorm we propose thefollowing operation laws for linguistic variables
Definition 10 Considering any two linguistic terms 119904120572 119904120573isin
119878 = 119904120574| 1199040lt 119904120574le 119904119905 120574 isin [0 119905] the add and multiply
operations of linguistic variable are defined as follows
119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905119878(120572119905120573119905)
(8)
119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905119879(120572119905120573119905)
(9)
where 119878(120572119905 120573119905) and 119879(120572119905 120573119905) are 119905-conorm and 119905-normrespectively
Since 119878(120572119905 120573119905) 119879(120572119905 120573119905) isin [0 1] we have 119905119878(120572119905
120573119905) 119905119879(120572119905 120573119905) isin [0 119905] which indicate the operationresults match the original linguistic term set 119878 that is119904119905119878(120572119905120573119905)
119904119905119879(120572119905120573119905)
isin 119878 In addition it is worth noting thatbecause of the monotonicity of 119905-conorm and 119905-norm thevalues of function 119905119878(120572119905 120573119905) and 119905119879(120572119905 120573119905) aremonoton-ically increasing with the increasing of 120572 and 120573 which meansthe operation results obtained by (8) and (9) are in accordwith our intuition
If we take the well-known 119878119875(120572119905 120573119905) and 119879
119875(120572119905 120573119905)
into (8) and (9) respectively then they can be rewritten asfollows
119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905(120572119905+120573119905minus120572120573119905
2)= 119904120572+120573minus120572120573119905
119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905(120572120573119905
2)= 119904120572120573119905
(10)
Example 11 Let 119878 = 119904120572| 1199040lt 119904120572le 1199046 120572 isin [0 6] Applying
(10) we have 1199043oplus 1199044= 1199043+4minus3times46
= 1199045 1199044oplus 1199045= 1199044+5minus4times56
=
1199045667
1199043otimes 1199044= 1199043times46
= 1199042 and 119904
4otimes 1199045= 1199044times56
= 1199043333
Thus we obtain 119904
3oplus 1199044lt 1199044oplus 1199045and 1199043otimes 1199044lt 1199044otimes 1199045 Such
results seem to be intuitive and can be easily accepted
Alternatively if we take the operation laws ofDefinition 3we have 119904
3oplus 1199044= 1199047notin 119878 119904
4otimes 1199045= 11990420
notin 119878 where thesubscripts of 119904
7and 119904
20are bigger than the cardinality of
linguistic term set 119878 In addition if we extend the discreteterm set 119878 to a continuous term set 119878
1015840
= 119904120572
| 120572 isin
[0 119902] [35] where 119902 (119902 gt 119905) is a sufficiently large positive
4 Journal of Applied Mathematics
integer there is an unavoidable question on how to definethe semantics for 119904
7and 11990420 Obviously 119904
7or 11990420has different
semantics in different linguistic term set 119878 with differentcardinalities As a result it is unrealistic to assign semanticsto a given linguistic value 119904
120572derived from linguistic term set
with variable cardinality If we follow the method of 119904120572oplus 119904120573=
min119904120572+120573
119904119905 and 119904
120572otimes 119904120573= min119904
120572120573 119904119905 [36] for 119904
120572119904120573isin 119878 =
119904120574
| 1199040
lt 119904120574
le 119904119905 120574 isin [0 119905] then we have 119904
3oplus 1199044
=
1199044oplus 1199045= 1199046and 1199043otimes 1199044= 1199044otimes 1199045= 1199046 Such results seem to be
counter-intuitive and may not be easily accepted Applying(10) we can overcome the limitations that the subscripts ofthe linguistic variable are bigger than the cardinality of thecorresponding linguistic term set 119878 and obtain results agreedwith our intuition
Based on (10) we can get the following operation laws forLIFVs
Definition 12 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with
120582 gt 0 then
119860 oplus 119861 = (119904119905(120579119905+120583119905minus120579120583119905
2) 119904119905(120590]1199052)) = (119904
120579+120583minus120579120583119905 119904120590]119905) (11)
119860 otimes 119861 = (119904119905(120579120583119905
2) 119904119905(120590119905+]119905minus120590]1199052)) = (119904
120579120583119905 119904120590+]minus120590]119905) (12)
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582 ) (13)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) (14)
Some special cases of 120582119860 and 119860120582 are obtained as follows
If 119860 = (119904120579 119904120590) = (119904119905 1199040) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905(1minus(1minus119905119905)
120582) 119904119905(0119905)
120582) = (119904119905 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905(119905119905)120582 119904119905(1minus(1minus0119905)
120582)) = (119904
119905 1199040)
(15)
If 119860 = (119904120579 119904120590) = (1199040 119904119905) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905(1minus(1minus0119905)
120582) 119904119905(119905119905)120582) = (119904
0 119904119905)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905(0119905)120582 119904119905(1minus(1minus119905119905)
120582)) = (119904
0 119904119905)
(16)
If 119860 = (119904120579 119904120590) = (1199040 1199040) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582)=(119904119905(1minus(1minus0119905)
120582) 119904119905(0119905)
120582)=(1199040 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582))=(119904119905(0119905)
120582 119904119905(1minus(1minus0119905)
120582))=(1199040 1199040)
(17)
If 120582 rarr 0 then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (1199040 119904119905)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905 1199040)
(18)
If 120582 rarr +infin then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
0 119904119905)
(19)
Theorem 13 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with
120582 1205821 1205822gt 0 Then one has
(1) 120582(119860 oplus 119861) = 120582119860 oplus 120582119861
(2) 1205821119860 oplus 120582
2119860 = (120582
1+ 1205822)119860
(3) 119860120582 otimes 119861120582= (119860 otimes 119861)
120582
(4) 1198601205821 otimes 1198601205822 = 1198601205821+1205822
Proof (1) By (11) we have 119860 oplus 119861 = (119904120579+120583minus120579120583119905
119904120590]119905) Thus
based on (13) we have
120582 (119860 oplus 119861) = (119904119905(1minus(1minus(120579+120583minus120579120583119905)119905)
120582) 119904119905((120590]119905)119905)120582)
= (119904119905(1minus(1minus120579119905)
120582(1minus120583119905)
120582) 119904119905(120590119905)
120582(]119905)120582)
(20)
Similarly since 120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) and 120582119861 =
(119904119905(1minus(1minus120583119905)
120582) 119904119905(]119905)120582) then
120582119860 oplus 120582119861
= (119904119905(1minus(1minus120579119905)
120582)+119905(1minus(1minus120583119905)
120582)minus119905(1minus(1minus120579119905)
120582)(1minus(1minus120583119905)
120582)
119904119905(120590119905)
120582119905(]119905)120582119905)
= (119904119905(1minus(1minus120579119905)
120582(1minus120583119905)
120582) 119904119905(120590119905)
120582(]119905)120582)
(21)
Hence we obtain 120582(119860 oplus 119861) = 120582119860 oplus 120582119861(2) By (13) we have 120582
1119860 = (119904
119905(1minus(1minus120579119905)1205821 ) 119904119905(120590119905)
1205821 ) and1205822119860 = (119904
119905(1minus(1minus120579119905)1205822 ) 119904119905(120590119905)
1205822 ) thus we obtain
1205821119860 oplus 120582
2119860
= (119904119905(1minus(1minus120579119905)
1205821 )+119905(1minus(1minus120579119905)1205822 )minus119905(1minus(1minus120579119905)
1205821 )(1minus(1minus120579119905)1205822 )
119904119905(120590119905)
1205821 (120590119905)1205822 )
= (119904119905(1minus120579119905)
1205821 (1minus120579119905)1205822 119904119905(120590119905)
1205821+1205822 )
= (119904119905(1minus120579119905)
1205821+1205822 119904119905(120590119905)1205821+1205822 ) = (120582
1+ 1205822) 119860
(22)
(3) By (14) we get 119860120582 = (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) and 119861
120582=
(119904119905(120583119905)
120582 119904119905(1minus(1minus]119905)120582)) thus based on (12) we have
119860120582otimes 119861120582
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582)+119905(1minus(1minus]119905)120582)minus119905(1minus(1minus120590119905)120582)(1minus(1minus]119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(23)
Journal of Applied Mathematics 5
Since 119860 otimes 119861 = (119904120579120583119905
119904120590+]minus120590]119905) then by (14) we have
(119860 otimes 119861)120582
= (119904119905(120579120583119905119905)
120582 119904119905(1minus(1minus(120590+]minus120590]119905)119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905minus]119905minus120590]1199052)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(24)
Hence we obtain 119860120582otimes 119861120582= (119860 otimes 119861)
120582(4) By (14) we have 119860
1205821 = (119904
119905(120579119905)1205821 119904119905(1minus(1minus120590119905)
1205821 )) and
1198601205822 = (119904119905(120579119905)
1205822 119904119905(1minus(1minus120590119905)1205822 )) thus we have
1198601205821otimes 1198601205822
= (119904119905(120579119905)
1205821 (120579119905)1205822
119904119905(1minus(1minus120590119905)
1205821 )+119905(1minus(1minus120590119905)1205822 )minus119905(1minus(1minus120590119905)
1205821 )(1minus(1minus120590119905)1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821 )(1minus120590119905)
1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )
) = 1198601205821+1205822
(25)
which completes the proof of Theorem 13
Motivated by the intuitionistic fuzzy aggregation oper-ators [14 15] in what follows we define some aggregationoperators for LIFVs
Definition 14 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedaveraging (LIFWA) operator is defined as
LIFWA (1198601 1198602 119860
119899) = 11990811198601oplus 11990821198602oplus sdot sdot sdot oplus 119908
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 )
(26)
where 119908 = (1199081 1199082 119908
119899)119879 is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
In Particular if 119908 = (1119899 1119899 1119899)119879 then the
LIFWA operator is reduced to a linguistic intuitionistic fuzzyaveraging (LIFA) operator that is
LIFWA (1198601 1198602 119860
119899) =
1
1198991198601oplus1
1198991198602oplus sdot sdot sdot oplus
1
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(27)
Based on Definition 14 we get some properties of theLIFWA operator
Theorem 15 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
0 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has
the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (28)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of LIFVs If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119860(119860
1 1198602 119860
119899)
(29)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(30)
Proof (1) Since 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
LIFWA (1198601 1198602 119860
119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579119905)
119908119894 ) 119904119905prod119899
119894=1(120590119905)119908119894 )
= (119904119905(1minus(1minus120579119905))
119904119905(120590119905)
) = (119904120579 119904120590)
(31)
(2) If 1199041205791015840
119894
ge 119904120579119894
that is 1205791015840119894ge 120579119894 for any 119894 then we have
1205791015840
119894ge 120579119894997904rArr 0 le 1 minus
1205791015840
119894
119905le 1 minus
120579119894
119905le 1
997904rArr (1 minus1205791015840
119894
119905)
119908119894
le (1 minus120579119894
119905)
119908119894
997904rArr
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
le
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 1 minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
ge 1 minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 119905(1minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
) ge 119905(1minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
)
(32)
Similarly when 1199041205901015840
119894
le 119904120590119894
for any 119894 we can get119905prod119899
119894=1(1205901015840
119894119905)1119899
le 119905prod119899
119894=1(120590119894119905)1119899
According to Definition 7 we obtain
(119904119905(1minusprod
119899
119894=1(1minus1205791015840
119894119905)1119899) 119904119905prod119899
119894=1(1205901015840
119894119905)1119899)
ge (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(33)
that is
LIFWA (1198601015840
1 1198601015840
2 119860
1015840
119899) ge LIFWA (119860
1 1198602 119860
119899)
(34)
6 Journal of Applied Mathematics
(3) Since min119894(119904120579119894
) le 119904120579119894
le max119894(119904120579119894
) and max119894(119904120590119894
) le
119904120590119894
le min119894(119904120590119894
) for any 119894 then based on the monotonicityof Theorem 15 we derive
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le LIFWA (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(35)
Definition 16 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted averaging (LIFOWA) operator is defined as
LIFOWA (1198601 1198602 119860
119899)
= 1199081119860(1)
oplus 1199082119860(2)
oplus sdot sdot sdot oplus 119908119899119860(119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899
119894=1(120590(119894)119905)119908119894 )
(36)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWA operator
Theorem 17 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set oflinguistic intuitionistic fuzzy values and119908 = (119908
1 1199082 119908
119899)119879
the associated weight vector of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin
[0 1] and sum119899119894=1
119908119894= 1 then one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (37)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(38)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(39)
(4) Commutativity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 0 1 2 119899)
be a set of linguistic intuitionistic fuzzy values then forany 119908
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) = 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(40)
where (1198601015840
1 1198601015840
2 119860
1015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Definition 18 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedgeometric (LIFWG) operator is defined as
LIFWG (1198601 1198602 119860
119899) = 1198601199081
1otimes 1198601199082
2otimes sdot sdot sdot otimes 119860
119908119899
119899
= (119904119905prod119899
119894=1(120579119894119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 ))
(41)
where 119908 = (1199081 1199082 119908
119899)119879is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
The LIFWG operator has the following properties
Theorem 19 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has the
following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119866 (1198601 1198602 119860
119899) = (119904
120579 119904120590) (42)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119866(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
(43)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119866 (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(44)
Definition 20 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted geometric (LIFOWG) operator is defined as
LIFOWG(1198601 1198602 119860
119899)
= 1198601199081
(1)otimes 1198601199082
(2)otimes sdot sdot sdot otimes 119860
119908119899
(119899)
= (119904119905prod119899
119894=1(120579(119894)119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590(119894)119905)119908119894 ))
(45)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWG operator
Journal of Applied Mathematics 7
Theorem 21 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119866(1198601 1198602 119860
119899) = (119904
120579 119904120590) (46)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(47)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119866(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(48)
(4) Commutativity Let1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values then for any119908
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(49)
where (1198601015840
1 1198601015840
2 1198601015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Lemma 22 (see [37 38]) Let 119909119894gt 0 120582
119894gt 0 119894 = 1 2 119899
and sum119899119894=1
120582119894= 1 then
119899
prod
119894=1
119909120582119894
119894le
119899
sum
119894=1
120582119894119909119894 (50)
with equality if and only if 1199091= 1199092= sdot sdot sdot = 119909
119899
Based on Lemma 22 we have the following theorem
Theorem 23 Let 119860119894= (119904120579119894
119904120590119894
)(119894 = 1 2 119899) be a set ofLIFVs then one has
119871119868119865119882119860(1198601 1198602 119860
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
119871119868119865119874119882119860(1198601 1198602 119860
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(51)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Proof Let 119908 = (1199081 1199082 119908
119899)119879 be the weight vector of
119860119894(119894 = 1 2 119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
by Lemma 22 we have 1 minusprod119899
119894=1(1 minus 120579
119894119905)119908119894 ge 1 minus sum
119899
119894=1119908119894(1 minus
120579119894119905) = 1minussum
119899
119894=1119908119894+sum119899
119894=1119908119894120579119894119905 = sum
119899
119894=1119908119894120579119894119905 ge prod
119899
119894=1(120579119894119905)119908119894
with equality if and only if 1205791
= 1205792
= sdot sdot sdot = 120579119899 that is
119905(1 minus prod119899
119894=1(1 minus 120579
119894119905)119908119894) ge 119905prod
119899
119894=1(120579119894119905)119908119894 with equality if and
only if 1205791= 1205792= sdot sdot sdot = 120579
119899 and prod
119899
119894=1(120590119894119905)119908119894 le sum
119899
119894=1119908119894120590119894119905 =
1 minus sum119899
119894=1119908119894(1 minus 120590
119894119905) le 1 minus prod
119899
119894=1(1 minus 120590
119894119905)119908119894 with equality if
and only if 1205901= 1205902= sdot sdot sdot = 120590
119899 that is 119905(prod119899
119894=1(120590119894119905)119908119894) le
119905(1 minus prod119899
119894=1(1 minus 120590
119894119905)119908119894) with equality if and only if 120590
1= 1205902=
sdot sdot sdot = 120590119899
Consequently by Definition 7 we obtain(119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 ) ge (119904
119905prod119899
119894=1(120579119894119905)119908119894
119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 )) with equality if and only if 120579
1= 1205792
=
sdot sdot sdot = 120579119899and 120590
1= 1205902= sdot sdot sdot = 120590
119899 that is
LIFWA (1198601 1198602 119860
119899) ge LIFWG (119860
1 1198602 119860
119899) (52)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Similarly we can also prove LIFOWA(1198601 1198602 119860
119899) ge
LIFOWG(1198601 1198602 119860
119899) with equality if and only if 119860
1=
1198602= sdot sdot sdot = 119860
119899
Besides the above properties we can derive the followingdesirable results of the LIFOWAandLIFOWGoperators
Theorem 24 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899
119894=1119908119894= 1 Then
one has the following
(1) If 119908 = (1 0 0)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= max119894
(119860119894)
(53)
(2) If 119908 = (0 0 1)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= min119894
(119860119894)
(54)
(3) If 119908119894= 1 and 119908
119895= 0 for 119895 = 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= 119860(119894)
(55)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of1198601 1198602 119860
119899
Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)
and 1198604= (1199044 1199041) be LIFVs which are derived from 119878 = 119904
120572|
1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)
119879 be theweight vector of 119860
119894(119894 = 1 2 3 4)
8 Journal of Applied Mathematics
Applying (26) and (41) we obtain the aggregated LIFVsas follows
LIFWA (1198601 1198602 119860
4) = (119904
6(1minusprod4
119894=1(1minus1205791198946)119908119894 ) 1199046prod4
119894=1(1205901198946)119908119894 )
= (1199041827
1199042781
)
LIFWG (1198601 1198602 119860
4) = (119904
6prod4
119894=1(1205791198946)119908119894 1199046(1minusprod
4
119894=1(1minus1205901198946)119908119894 ))
= (1199041516
1199043157
)
(56)
It is easy to see that
LIFWA (1198601 1198602 119860
4) = (119904
1827 1199042781
)
gt LIFWG (1198601 1198602 119860
4)
= (1199041516
1199043157
)
(57)
By Definition 7 we calculate the following values of scorefunction and accuracy function
119878 (1198601) = 119904(6+1minus2)2
= 11990425 119878 (119860
2) = 119904(6+1minus3)2
= 1199042
119878 (1198603) = 119904(6+2minus4)2
= 1199042 119878 (119860
4) = 119904(6+4minus1)2
= 11990445
119867 (1198602) = 1199041+3
= 1199044 119867 (119860
3) = 1199042+4
= 1199046
(58)
Since 119878(1198604) gt 119878(119860
1) gt 119878(119860
2) = 119878(119860
3) and 119867(119860
3) gt
119867(1198602) then
119860(1)
= 1198604= (1199044 1199041) 119860
(2)= 1198601= (1199041 1199042)
119860(3)
= 1198603= (1199042 1199044) 119860
(4)= 1198602= (1199041 1199043)
(59)
Assume that 119908 = (0155 0345 0345 0155)119879 which are
determined by the normal distribution based method [39] isthe associated weight vector of 119860
(119894) Then by (36) and (45)
we have
LIFOWA (1198601 1198602 119860
119899)
= (1199046(1minusprod
4
119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899
119894=1(120590(119894)6)119908119894)
= (1199041983
1199042430
)
LIFOWG (1198601 1198602 119860
119899)
= (1199046prod119899
119894=1(120579(119894)6)119908119894 1199046(1minusprod
119899
119894=1(1minus120590(119894)6)119908119894 ))
= (1199041575
1199042882
)
(60)
It is easy to see that
LIFOWA (1198601 1198602 119860
4) = (119904
1983 1199042430
)
gt LIFOWG (1198601 1198602 119860
4)
= (1199041575
1199042882
)
(61)
5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information
In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs
Let 119860 = (1198601 1198602 119860
119898) be the set of alternatives and
119862 = (1198621 1198622 119862
119899) the set of attributes whose weight vector
is 119908 = (1199081 1199082 119908
119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 Let
119863 = (1198631 1198632 119863
119897) be the set of decision makers Suppose
119877119896= (119903119896
119894119895)119898times119899
is the decision matrix where 119903119896
119894119895= (119904119896
120579119894119895
119904119896
120590119894119895
)
denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863
119896for alternative 119860
119894with
respect to attribute 119862119895and 119904119896
120579119894119895
119904119896
120590119894119895
isin 119878 = 119904119894| 119894 = 0 1 119905
The proposed method is described as follows
Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877
119896= (119903119896
119894119895)119898times119899
Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903
119894119895)119898times119899
119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWA (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
or 119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWG (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
(62)
where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]
Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective
overall preference values 119903119894for each alternative 119860
119894(119894 =
1 2 119898) based on the LIFWA or LIFWG operator
Step 4 Rank the alternatives in accordance with 119903119894 according
to Definition 7
6 Numerical Example
In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860
1is a car industry 119860
2
is a food company 1198603is a computer company 119860
4is
an arms industry The investment company must make adecision according to four criteria 119862
1is the risk analysis
1198622is the growth analysis 119862
3is the social-political impact
analysis 1198624is the environmental impact analysis The weight
vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =
1199040
= extremely poor 1199041
= very poor 1199042
= poor 1199043
=
slightly poor 1199044= fair 119904
5= slightly good 119904
6= good 119904
7=
very good 1199048= extremely good
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
2 Preliminaries
In the following some basic concepts and knowledge relatedto IFS and linguistic approach are briefly described
Definition 1 (see [1]) Let119883 be a universal set An IFS 119860 in119883
is given as
119860 = (119909 120583119860 (119909) V119860 (119909)) | 119909 isin 119883 (1)
where the functions 120583119860(119909) 119883 rarr [0 1] V
119860(119909) 119883 rarr [0 1]
stand for the degree of membership and nonmembershipof the element 119909 to 119860 respectively Any 119909 isin 119883 meets thecondition 0 le 120583
119860(119909) + V
119860(119909) le 1
120587119860(119909) is called intuitionistic index or degree of indeter-
minacy of 119909 to 119860 120587119860(119909) = 1 minus 120583
119860(119909) minus V
119860(119909) Obviously if
120587119860(119909) = 0 IFS 119860 is reduced to a fuzzy setSome basic definitions and operations on IFS are pre-
sented as follows
Definition 2 (see [14 15]) If 119860 and 119861 are two IFSs of the set119883 then
(1) 119860 = 119861 if and only if forall119909 isin 119883 120583119860(119909) = 120583
119861(119909) and
V119860(119909) = V
119861(119909)
(2) 119860119888 = (119909 V119860(119909) 120583119860(119909)) | 119909 isin 119883 where 119860
119888 is thecomplement of 119860
(3) 119860 and 119861 = (min(120583119860(119909) 120583119861(119909))max(V
119860(119909) V119861(119909)))
(4) 119860 or 119861 = (max(120583119860(119909) 120583119861(119909))min(V
119860(119909) V119861(119909)))
(5) 119860 oplus 119861 = (120583119860(119909) + 120583
119861(119909) minus 120583
119860(119909)120583119861(119909) V119860(119909)V119861(119909))
(6) 119860 otimes 119861 = (120583119860(119909)120583119861(119909) V119860(119909) + V
119861(119909) minus V
119860(119909)V119861(119909))
(7) 120582119860 = (1 minus (1 minus 120583119860(119909))120582 V119860(119909)120582) 120582 gt 0
(8) 119860120582 = (120583119860(119909)120582 1 minus (1 minus V
119860(119909))120582) 120582 gt 0
In real world many decision making problems presentqualitative aspects that are complex to assess by means ofnumerical values In such cases it may be more suitable toconsider them as linguistic variables
Let 119878 = 119904119894| 119894 = 0 1 119905 be a finite linguistic term set
with odd cardinality where 119904119894represents a possible linguistic
term for a linguistic variable For example a set of seven terms119878 can be expressed as follows
119878 = 1199040= N (none) 1199041 = VL (very low) 119904
2= L (low)
1199043= M (medium) 1199044 = H (high)
1199045= VH (very high) 119904
6= P (perfect)
(2)
It is required that the linguistic term set should satisfy thefollowing characteristics [27ndash30]
(1) The set is ordered 119904119894gt 119904119895 if and only if 119894 gt 119895
(2) There is a negation operator Neg(119904119894) = 119904119895such that
119895 = 119905 minus 119894(3) Max operator max(119904
119894 119904119895) = 119904119894 if and only if 119894 ge 119895
(4) Min operator min(119904119894 119904119895) = 119904119894 if and only if 119894 le 119895
To preserve all the given information Xu [31] extendedthe discrete term set 119878 to a continuous linguistic term set 119878 =
119904120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] where if 119904
120572isin 119878 then 119904
120572is called
the original linguistic term Otherwise 119904120572is called the virtual
linguistic term
Definition 3 (see [31 32]) Consider any two linguistic terms119904120572 119904120573
isin 119878 and 120583 1205831 1205832
isin [0 1] the add and multiplyoperations of linguistic variable are defined as follows
119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904120572+120573
119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904120572120573
(3)
119905-norm and 119905-conorm have been widely used to constructoperations for fuzzy sets and IFSs
Definition 4 (see [33 34]) A 119905-norm is a mapping 119879 [0 1] times
[0 1] rarr [0 1] satisfying for all 119909 119910 119911 isin [0 1]
(1) 119879(119909 1) = 119909
(2) 119879(119909 119910) = 119879(119910 119909)
(3) 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)
(4) 119879(119909 119910) le 119879(119909 119911) whenever 119910 le 119911
The four basic 119905-norms 119879119872 119879119875 119879119871 and 119879
119863are given as
follows
119879119872(119909 119910) = min(119909 119910) (lattice operation)
119879119875(119909 119910) = 119909 sdot 119910 (algebraic operation)
119879119871(119909 119910) = max(119909+119910minus1 0) (Lukasiewicz operation)
119879119863(119909 119910) =
0 if (119909119910)isin[01)2min(119909119910) otherwise (drastic operation)
Definition 5 (see [33 34]) A t-conorm is a mapping 119878
[0 1] times [0 1] rarr [0 1] satisfying for all 119909 119910 119911 isin [0 1]
(1) 119878(119909 0) = 119909
(2) 119878(119909 119910) = 119878(119910 119909)
(3) 119878(119909 119878(119910 119911)) = 119878(119878(119909 119910) 119911)
(4) 119878(119909 119910) le 119878(119909 119911) whenever 119910 le 119911
The four basic 119905-conorms 119878119872 119878119875 119878119871 and 119878
119863are given as
follows
119878119872(119909 119910) = max(119909 119910) (lattice operation)
119878119875(119909 119910) = 119909 + 119910 minus 119909 sdot 119910 (algebraic operation)
119878119871(119909 119910) = min(119909 + 119910 1) (Lukasiewicz operation)
119878119863(119909 119910) =
max(119909119910) if (119909119910)isin(01]21 otherwise
(drastic operation)
3 Linguistic Intuitionistic Fuzzy Set
The concept of linguistic intuitionistic fuzzy set (LIFS) isgiven as follows
Journal of Applied Mathematics 3
Definition 6 Let119883 be a finite universal set and 119878 = 119904120572| 1199040lt
119904120572le 119904119905 120572 isin [0 119905] a continuous linguistic term set A LIFS 119860
in119883 is given as
119860 = (119909 119904120579 (119909) 119904120590 (119909)) | 119909 isin 119883 (4)
where 119904120579(119909) 119904120590(119909) isin 119878 stand for the linguistic membership
degree and linguistic nonmembership of the element 119909 to 119860respectively
For any 119909 isin 119883 the condition 0 le 120579 + 120590 le 119905 is alwayssatisfied 120587(119909) is called linguistic indeterminacy degree of 119909to 119860 120587(119909) = 119904
119905minus120579minus120590 Obviously if 120579 + 120590 = 119905 then LIFS
119860 has the minimum linguistic indeterminacy degree thatis 120587(119909) = 119904
0 which means the membership degree of 119909 to
119860 can be precisely expressed with a single linguistic termand LIFS 119860 is reduced to a linguistic variable Oppositelyif 120579 = 120590 = 0 then LIFS 119860(119909) has the maximum linguisticindeterminacy degree that is 120587(119909) = 119904
119905 Similar to IFS
the LIFS 119860(119909) can be transformed into an interval linguisticvariable [119904
120579(119909) 119904119905minus120590
(119909)] which indicates that the minimumandmaximum linguisticmembership degrees of the elements119909 to 119860 are 119904
120579and 119904119905minus120590
respectivelyFor notational simplicity we suppose both LIFS 119860 and
119861 contain only one element which stand for linguisticintuitionistic fuzzy values (LIFVs) that is the pairs 119860 =
(119904120579 119904120590) and 119861 = (119904
120583 119904])
To compare any two LIFVs the score function andaccuracy function are defined as follows
Definition 7 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
with 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] The score
function of 119860 is defined as
119878 (119860) = 119904(119905+120579minus120590)2
(5)
and the accuracy function is defined as
119867(119860) = 119904120579+120590
(6)
Thus 119860 and 119861 can be ranked by the following procedure
(1) if 119878(119860) gt 119878(119861) then 119860 gt 119861(2) if 119878(119860) = 119878(119861) and
(a) 119867(119860) = 119867(119861) then 119860 = 119861(b) 119867(119860) gt 119867(119861) then 119860 gt 119861(c) 119867(119860) lt 119867(119861) then 119860 lt 119861
It is easy to see that 0 le (119905+120579minus120590)2 le 119905 and 0 le 120579+120590 le 119905which means 119904
(119905+120579minus120590)2 119904120579+120590
isin 119878
Example 8 Let 119860 = (1199041 1199043) 119861 = (119904
0 1199044) and 119862 = (119904
1 1199045) be
LIFVs which are derived from 119878 = 119904120572| 1199040lt 119904120572le 1199046 120572 isin
[0 6]Applying formulas (5) and (6) we have
119878 (119860) = 1199042gt 119878 (119861) = 119878 (119862) = 119904
1
119867 (119861) = 1199044lt 119867 (119862) = 119904
6
(7)
Thus we obtain 119860 gt 119862 gt 119861
4 Aggregation Operators for LinguisticIntuitionistic Fuzzy Sets
Since the definition of LIFS is given it is necessary tointroduce the operations and computations between them
Definition 9 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
then
(1) 119860 = 119861 if and only if 119904120579= 119904120583and 119904120590= 119904]
(2) 119860119888 = (119904120590 119904120579) where 119860119888 is the complement of 119860
(3) the intersection of 119860 and 119861 119860 and 119861 =
(min(119904120579 119904120583)max(119904
120590 119904]))
(4) the union of 119860 and 119861 119860 or 119861 =
(max(119904120579 119904120583)min(119904
120590 119904]))
Motivated by 119905-norm and 119905-conorm we propose thefollowing operation laws for linguistic variables
Definition 10 Considering any two linguistic terms 119904120572 119904120573isin
119878 = 119904120574| 1199040lt 119904120574le 119904119905 120574 isin [0 119905] the add and multiply
operations of linguistic variable are defined as follows
119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905119878(120572119905120573119905)
(8)
119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905119879(120572119905120573119905)
(9)
where 119878(120572119905 120573119905) and 119879(120572119905 120573119905) are 119905-conorm and 119905-normrespectively
Since 119878(120572119905 120573119905) 119879(120572119905 120573119905) isin [0 1] we have 119905119878(120572119905
120573119905) 119905119879(120572119905 120573119905) isin [0 119905] which indicate the operationresults match the original linguistic term set 119878 that is119904119905119878(120572119905120573119905)
119904119905119879(120572119905120573119905)
isin 119878 In addition it is worth noting thatbecause of the monotonicity of 119905-conorm and 119905-norm thevalues of function 119905119878(120572119905 120573119905) and 119905119879(120572119905 120573119905) aremonoton-ically increasing with the increasing of 120572 and 120573 which meansthe operation results obtained by (8) and (9) are in accordwith our intuition
If we take the well-known 119878119875(120572119905 120573119905) and 119879
119875(120572119905 120573119905)
into (8) and (9) respectively then they can be rewritten asfollows
119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905(120572119905+120573119905minus120572120573119905
2)= 119904120572+120573minus120572120573119905
119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905(120572120573119905
2)= 119904120572120573119905
(10)
Example 11 Let 119878 = 119904120572| 1199040lt 119904120572le 1199046 120572 isin [0 6] Applying
(10) we have 1199043oplus 1199044= 1199043+4minus3times46
= 1199045 1199044oplus 1199045= 1199044+5minus4times56
=
1199045667
1199043otimes 1199044= 1199043times46
= 1199042 and 119904
4otimes 1199045= 1199044times56
= 1199043333
Thus we obtain 119904
3oplus 1199044lt 1199044oplus 1199045and 1199043otimes 1199044lt 1199044otimes 1199045 Such
results seem to be intuitive and can be easily accepted
Alternatively if we take the operation laws ofDefinition 3we have 119904
3oplus 1199044= 1199047notin 119878 119904
4otimes 1199045= 11990420
notin 119878 where thesubscripts of 119904
7and 119904
20are bigger than the cardinality of
linguistic term set 119878 In addition if we extend the discreteterm set 119878 to a continuous term set 119878
1015840
= 119904120572
| 120572 isin
[0 119902] [35] where 119902 (119902 gt 119905) is a sufficiently large positive
4 Journal of Applied Mathematics
integer there is an unavoidable question on how to definethe semantics for 119904
7and 11990420 Obviously 119904
7or 11990420has different
semantics in different linguistic term set 119878 with differentcardinalities As a result it is unrealistic to assign semanticsto a given linguistic value 119904
120572derived from linguistic term set
with variable cardinality If we follow the method of 119904120572oplus 119904120573=
min119904120572+120573
119904119905 and 119904
120572otimes 119904120573= min119904
120572120573 119904119905 [36] for 119904
120572119904120573isin 119878 =
119904120574
| 1199040
lt 119904120574
le 119904119905 120574 isin [0 119905] then we have 119904
3oplus 1199044
=
1199044oplus 1199045= 1199046and 1199043otimes 1199044= 1199044otimes 1199045= 1199046 Such results seem to be
counter-intuitive and may not be easily accepted Applying(10) we can overcome the limitations that the subscripts ofthe linguistic variable are bigger than the cardinality of thecorresponding linguistic term set 119878 and obtain results agreedwith our intuition
Based on (10) we can get the following operation laws forLIFVs
Definition 12 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with
120582 gt 0 then
119860 oplus 119861 = (119904119905(120579119905+120583119905minus120579120583119905
2) 119904119905(120590]1199052)) = (119904
120579+120583minus120579120583119905 119904120590]119905) (11)
119860 otimes 119861 = (119904119905(120579120583119905
2) 119904119905(120590119905+]119905minus120590]1199052)) = (119904
120579120583119905 119904120590+]minus120590]119905) (12)
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582 ) (13)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) (14)
Some special cases of 120582119860 and 119860120582 are obtained as follows
If 119860 = (119904120579 119904120590) = (119904119905 1199040) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905(1minus(1minus119905119905)
120582) 119904119905(0119905)
120582) = (119904119905 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905(119905119905)120582 119904119905(1minus(1minus0119905)
120582)) = (119904
119905 1199040)
(15)
If 119860 = (119904120579 119904120590) = (1199040 119904119905) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905(1minus(1minus0119905)
120582) 119904119905(119905119905)120582) = (119904
0 119904119905)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905(0119905)120582 119904119905(1minus(1minus119905119905)
120582)) = (119904
0 119904119905)
(16)
If 119860 = (119904120579 119904120590) = (1199040 1199040) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582)=(119904119905(1minus(1minus0119905)
120582) 119904119905(0119905)
120582)=(1199040 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582))=(119904119905(0119905)
120582 119904119905(1minus(1minus0119905)
120582))=(1199040 1199040)
(17)
If 120582 rarr 0 then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (1199040 119904119905)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905 1199040)
(18)
If 120582 rarr +infin then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
0 119904119905)
(19)
Theorem 13 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with
120582 1205821 1205822gt 0 Then one has
(1) 120582(119860 oplus 119861) = 120582119860 oplus 120582119861
(2) 1205821119860 oplus 120582
2119860 = (120582
1+ 1205822)119860
(3) 119860120582 otimes 119861120582= (119860 otimes 119861)
120582
(4) 1198601205821 otimes 1198601205822 = 1198601205821+1205822
Proof (1) By (11) we have 119860 oplus 119861 = (119904120579+120583minus120579120583119905
119904120590]119905) Thus
based on (13) we have
120582 (119860 oplus 119861) = (119904119905(1minus(1minus(120579+120583minus120579120583119905)119905)
120582) 119904119905((120590]119905)119905)120582)
= (119904119905(1minus(1minus120579119905)
120582(1minus120583119905)
120582) 119904119905(120590119905)
120582(]119905)120582)
(20)
Similarly since 120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) and 120582119861 =
(119904119905(1minus(1minus120583119905)
120582) 119904119905(]119905)120582) then
120582119860 oplus 120582119861
= (119904119905(1minus(1minus120579119905)
120582)+119905(1minus(1minus120583119905)
120582)minus119905(1minus(1minus120579119905)
120582)(1minus(1minus120583119905)
120582)
119904119905(120590119905)
120582119905(]119905)120582119905)
= (119904119905(1minus(1minus120579119905)
120582(1minus120583119905)
120582) 119904119905(120590119905)
120582(]119905)120582)
(21)
Hence we obtain 120582(119860 oplus 119861) = 120582119860 oplus 120582119861(2) By (13) we have 120582
1119860 = (119904
119905(1minus(1minus120579119905)1205821 ) 119904119905(120590119905)
1205821 ) and1205822119860 = (119904
119905(1minus(1minus120579119905)1205822 ) 119904119905(120590119905)
1205822 ) thus we obtain
1205821119860 oplus 120582
2119860
= (119904119905(1minus(1minus120579119905)
1205821 )+119905(1minus(1minus120579119905)1205822 )minus119905(1minus(1minus120579119905)
1205821 )(1minus(1minus120579119905)1205822 )
119904119905(120590119905)
1205821 (120590119905)1205822 )
= (119904119905(1minus120579119905)
1205821 (1minus120579119905)1205822 119904119905(120590119905)
1205821+1205822 )
= (119904119905(1minus120579119905)
1205821+1205822 119904119905(120590119905)1205821+1205822 ) = (120582
1+ 1205822) 119860
(22)
(3) By (14) we get 119860120582 = (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) and 119861
120582=
(119904119905(120583119905)
120582 119904119905(1minus(1minus]119905)120582)) thus based on (12) we have
119860120582otimes 119861120582
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582)+119905(1minus(1minus]119905)120582)minus119905(1minus(1minus120590119905)120582)(1minus(1minus]119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(23)
Journal of Applied Mathematics 5
Since 119860 otimes 119861 = (119904120579120583119905
119904120590+]minus120590]119905) then by (14) we have
(119860 otimes 119861)120582
= (119904119905(120579120583119905119905)
120582 119904119905(1minus(1minus(120590+]minus120590]119905)119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905minus]119905minus120590]1199052)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(24)
Hence we obtain 119860120582otimes 119861120582= (119860 otimes 119861)
120582(4) By (14) we have 119860
1205821 = (119904
119905(120579119905)1205821 119904119905(1minus(1minus120590119905)
1205821 )) and
1198601205822 = (119904119905(120579119905)
1205822 119904119905(1minus(1minus120590119905)1205822 )) thus we have
1198601205821otimes 1198601205822
= (119904119905(120579119905)
1205821 (120579119905)1205822
119904119905(1minus(1minus120590119905)
1205821 )+119905(1minus(1minus120590119905)1205822 )minus119905(1minus(1minus120590119905)
1205821 )(1minus(1minus120590119905)1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821 )(1minus120590119905)
1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )
) = 1198601205821+1205822
(25)
which completes the proof of Theorem 13
Motivated by the intuitionistic fuzzy aggregation oper-ators [14 15] in what follows we define some aggregationoperators for LIFVs
Definition 14 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedaveraging (LIFWA) operator is defined as
LIFWA (1198601 1198602 119860
119899) = 11990811198601oplus 11990821198602oplus sdot sdot sdot oplus 119908
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 )
(26)
where 119908 = (1199081 1199082 119908
119899)119879 is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
In Particular if 119908 = (1119899 1119899 1119899)119879 then the
LIFWA operator is reduced to a linguistic intuitionistic fuzzyaveraging (LIFA) operator that is
LIFWA (1198601 1198602 119860
119899) =
1
1198991198601oplus1
1198991198602oplus sdot sdot sdot oplus
1
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(27)
Based on Definition 14 we get some properties of theLIFWA operator
Theorem 15 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
0 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has
the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (28)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of LIFVs If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119860(119860
1 1198602 119860
119899)
(29)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(30)
Proof (1) Since 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
LIFWA (1198601 1198602 119860
119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579119905)
119908119894 ) 119904119905prod119899
119894=1(120590119905)119908119894 )
= (119904119905(1minus(1minus120579119905))
119904119905(120590119905)
) = (119904120579 119904120590)
(31)
(2) If 1199041205791015840
119894
ge 119904120579119894
that is 1205791015840119894ge 120579119894 for any 119894 then we have
1205791015840
119894ge 120579119894997904rArr 0 le 1 minus
1205791015840
119894
119905le 1 minus
120579119894
119905le 1
997904rArr (1 minus1205791015840
119894
119905)
119908119894
le (1 minus120579119894
119905)
119908119894
997904rArr
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
le
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 1 minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
ge 1 minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 119905(1minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
) ge 119905(1minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
)
(32)
Similarly when 1199041205901015840
119894
le 119904120590119894
for any 119894 we can get119905prod119899
119894=1(1205901015840
119894119905)1119899
le 119905prod119899
119894=1(120590119894119905)1119899
According to Definition 7 we obtain
(119904119905(1minusprod
119899
119894=1(1minus1205791015840
119894119905)1119899) 119904119905prod119899
119894=1(1205901015840
119894119905)1119899)
ge (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(33)
that is
LIFWA (1198601015840
1 1198601015840
2 119860
1015840
119899) ge LIFWA (119860
1 1198602 119860
119899)
(34)
6 Journal of Applied Mathematics
(3) Since min119894(119904120579119894
) le 119904120579119894
le max119894(119904120579119894
) and max119894(119904120590119894
) le
119904120590119894
le min119894(119904120590119894
) for any 119894 then based on the monotonicityof Theorem 15 we derive
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le LIFWA (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(35)
Definition 16 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted averaging (LIFOWA) operator is defined as
LIFOWA (1198601 1198602 119860
119899)
= 1199081119860(1)
oplus 1199082119860(2)
oplus sdot sdot sdot oplus 119908119899119860(119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899
119894=1(120590(119894)119905)119908119894 )
(36)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWA operator
Theorem 17 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set oflinguistic intuitionistic fuzzy values and119908 = (119908
1 1199082 119908
119899)119879
the associated weight vector of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin
[0 1] and sum119899119894=1
119908119894= 1 then one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (37)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(38)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(39)
(4) Commutativity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 0 1 2 119899)
be a set of linguistic intuitionistic fuzzy values then forany 119908
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) = 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(40)
where (1198601015840
1 1198601015840
2 119860
1015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Definition 18 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedgeometric (LIFWG) operator is defined as
LIFWG (1198601 1198602 119860
119899) = 1198601199081
1otimes 1198601199082
2otimes sdot sdot sdot otimes 119860
119908119899
119899
= (119904119905prod119899
119894=1(120579119894119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 ))
(41)
where 119908 = (1199081 1199082 119908
119899)119879is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
The LIFWG operator has the following properties
Theorem 19 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has the
following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119866 (1198601 1198602 119860
119899) = (119904
120579 119904120590) (42)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119866(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
(43)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119866 (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(44)
Definition 20 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted geometric (LIFOWG) operator is defined as
LIFOWG(1198601 1198602 119860
119899)
= 1198601199081
(1)otimes 1198601199082
(2)otimes sdot sdot sdot otimes 119860
119908119899
(119899)
= (119904119905prod119899
119894=1(120579(119894)119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590(119894)119905)119908119894 ))
(45)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWG operator
Journal of Applied Mathematics 7
Theorem 21 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119866(1198601 1198602 119860
119899) = (119904
120579 119904120590) (46)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(47)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119866(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(48)
(4) Commutativity Let1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values then for any119908
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(49)
where (1198601015840
1 1198601015840
2 1198601015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Lemma 22 (see [37 38]) Let 119909119894gt 0 120582
119894gt 0 119894 = 1 2 119899
and sum119899119894=1
120582119894= 1 then
119899
prod
119894=1
119909120582119894
119894le
119899
sum
119894=1
120582119894119909119894 (50)
with equality if and only if 1199091= 1199092= sdot sdot sdot = 119909
119899
Based on Lemma 22 we have the following theorem
Theorem 23 Let 119860119894= (119904120579119894
119904120590119894
)(119894 = 1 2 119899) be a set ofLIFVs then one has
119871119868119865119882119860(1198601 1198602 119860
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
119871119868119865119874119882119860(1198601 1198602 119860
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(51)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Proof Let 119908 = (1199081 1199082 119908
119899)119879 be the weight vector of
119860119894(119894 = 1 2 119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
by Lemma 22 we have 1 minusprod119899
119894=1(1 minus 120579
119894119905)119908119894 ge 1 minus sum
119899
119894=1119908119894(1 minus
120579119894119905) = 1minussum
119899
119894=1119908119894+sum119899
119894=1119908119894120579119894119905 = sum
119899
119894=1119908119894120579119894119905 ge prod
119899
119894=1(120579119894119905)119908119894
with equality if and only if 1205791
= 1205792
= sdot sdot sdot = 120579119899 that is
119905(1 minus prod119899
119894=1(1 minus 120579
119894119905)119908119894) ge 119905prod
119899
119894=1(120579119894119905)119908119894 with equality if and
only if 1205791= 1205792= sdot sdot sdot = 120579
119899 and prod
119899
119894=1(120590119894119905)119908119894 le sum
119899
119894=1119908119894120590119894119905 =
1 minus sum119899
119894=1119908119894(1 minus 120590
119894119905) le 1 minus prod
119899
119894=1(1 minus 120590
119894119905)119908119894 with equality if
and only if 1205901= 1205902= sdot sdot sdot = 120590
119899 that is 119905(prod119899
119894=1(120590119894119905)119908119894) le
119905(1 minus prod119899
119894=1(1 minus 120590
119894119905)119908119894) with equality if and only if 120590
1= 1205902=
sdot sdot sdot = 120590119899
Consequently by Definition 7 we obtain(119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 ) ge (119904
119905prod119899
119894=1(120579119894119905)119908119894
119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 )) with equality if and only if 120579
1= 1205792
=
sdot sdot sdot = 120579119899and 120590
1= 1205902= sdot sdot sdot = 120590
119899 that is
LIFWA (1198601 1198602 119860
119899) ge LIFWG (119860
1 1198602 119860
119899) (52)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Similarly we can also prove LIFOWA(1198601 1198602 119860
119899) ge
LIFOWG(1198601 1198602 119860
119899) with equality if and only if 119860
1=
1198602= sdot sdot sdot = 119860
119899
Besides the above properties we can derive the followingdesirable results of the LIFOWAandLIFOWGoperators
Theorem 24 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899
119894=1119908119894= 1 Then
one has the following
(1) If 119908 = (1 0 0)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= max119894
(119860119894)
(53)
(2) If 119908 = (0 0 1)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= min119894
(119860119894)
(54)
(3) If 119908119894= 1 and 119908
119895= 0 for 119895 = 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= 119860(119894)
(55)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of1198601 1198602 119860
119899
Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)
and 1198604= (1199044 1199041) be LIFVs which are derived from 119878 = 119904
120572|
1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)
119879 be theweight vector of 119860
119894(119894 = 1 2 3 4)
8 Journal of Applied Mathematics
Applying (26) and (41) we obtain the aggregated LIFVsas follows
LIFWA (1198601 1198602 119860
4) = (119904
6(1minusprod4
119894=1(1minus1205791198946)119908119894 ) 1199046prod4
119894=1(1205901198946)119908119894 )
= (1199041827
1199042781
)
LIFWG (1198601 1198602 119860
4) = (119904
6prod4
119894=1(1205791198946)119908119894 1199046(1minusprod
4
119894=1(1minus1205901198946)119908119894 ))
= (1199041516
1199043157
)
(56)
It is easy to see that
LIFWA (1198601 1198602 119860
4) = (119904
1827 1199042781
)
gt LIFWG (1198601 1198602 119860
4)
= (1199041516
1199043157
)
(57)
By Definition 7 we calculate the following values of scorefunction and accuracy function
119878 (1198601) = 119904(6+1minus2)2
= 11990425 119878 (119860
2) = 119904(6+1minus3)2
= 1199042
119878 (1198603) = 119904(6+2minus4)2
= 1199042 119878 (119860
4) = 119904(6+4minus1)2
= 11990445
119867 (1198602) = 1199041+3
= 1199044 119867 (119860
3) = 1199042+4
= 1199046
(58)
Since 119878(1198604) gt 119878(119860
1) gt 119878(119860
2) = 119878(119860
3) and 119867(119860
3) gt
119867(1198602) then
119860(1)
= 1198604= (1199044 1199041) 119860
(2)= 1198601= (1199041 1199042)
119860(3)
= 1198603= (1199042 1199044) 119860
(4)= 1198602= (1199041 1199043)
(59)
Assume that 119908 = (0155 0345 0345 0155)119879 which are
determined by the normal distribution based method [39] isthe associated weight vector of 119860
(119894) Then by (36) and (45)
we have
LIFOWA (1198601 1198602 119860
119899)
= (1199046(1minusprod
4
119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899
119894=1(120590(119894)6)119908119894)
= (1199041983
1199042430
)
LIFOWG (1198601 1198602 119860
119899)
= (1199046prod119899
119894=1(120579(119894)6)119908119894 1199046(1minusprod
119899
119894=1(1minus120590(119894)6)119908119894 ))
= (1199041575
1199042882
)
(60)
It is easy to see that
LIFOWA (1198601 1198602 119860
4) = (119904
1983 1199042430
)
gt LIFOWG (1198601 1198602 119860
4)
= (1199041575
1199042882
)
(61)
5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information
In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs
Let 119860 = (1198601 1198602 119860
119898) be the set of alternatives and
119862 = (1198621 1198622 119862
119899) the set of attributes whose weight vector
is 119908 = (1199081 1199082 119908
119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 Let
119863 = (1198631 1198632 119863
119897) be the set of decision makers Suppose
119877119896= (119903119896
119894119895)119898times119899
is the decision matrix where 119903119896
119894119895= (119904119896
120579119894119895
119904119896
120590119894119895
)
denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863
119896for alternative 119860
119894with
respect to attribute 119862119895and 119904119896
120579119894119895
119904119896
120590119894119895
isin 119878 = 119904119894| 119894 = 0 1 119905
The proposed method is described as follows
Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877
119896= (119903119896
119894119895)119898times119899
Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903
119894119895)119898times119899
119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWA (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
or 119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWG (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
(62)
where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]
Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective
overall preference values 119903119894for each alternative 119860
119894(119894 =
1 2 119898) based on the LIFWA or LIFWG operator
Step 4 Rank the alternatives in accordance with 119903119894 according
to Definition 7
6 Numerical Example
In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860
1is a car industry 119860
2
is a food company 1198603is a computer company 119860
4is
an arms industry The investment company must make adecision according to four criteria 119862
1is the risk analysis
1198622is the growth analysis 119862
3is the social-political impact
analysis 1198624is the environmental impact analysis The weight
vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =
1199040
= extremely poor 1199041
= very poor 1199042
= poor 1199043
=
slightly poor 1199044= fair 119904
5= slightly good 119904
6= good 119904
7=
very good 1199048= extremely good
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Definition 6 Let119883 be a finite universal set and 119878 = 119904120572| 1199040lt
119904120572le 119904119905 120572 isin [0 119905] a continuous linguistic term set A LIFS 119860
in119883 is given as
119860 = (119909 119904120579 (119909) 119904120590 (119909)) | 119909 isin 119883 (4)
where 119904120579(119909) 119904120590(119909) isin 119878 stand for the linguistic membership
degree and linguistic nonmembership of the element 119909 to 119860respectively
For any 119909 isin 119883 the condition 0 le 120579 + 120590 le 119905 is alwayssatisfied 120587(119909) is called linguistic indeterminacy degree of 119909to 119860 120587(119909) = 119904
119905minus120579minus120590 Obviously if 120579 + 120590 = 119905 then LIFS
119860 has the minimum linguistic indeterminacy degree thatis 120587(119909) = 119904
0 which means the membership degree of 119909 to
119860 can be precisely expressed with a single linguistic termand LIFS 119860 is reduced to a linguistic variable Oppositelyif 120579 = 120590 = 0 then LIFS 119860(119909) has the maximum linguisticindeterminacy degree that is 120587(119909) = 119904
119905 Similar to IFS
the LIFS 119860(119909) can be transformed into an interval linguisticvariable [119904
120579(119909) 119904119905minus120590
(119909)] which indicates that the minimumandmaximum linguisticmembership degrees of the elements119909 to 119860 are 119904
120579and 119904119905minus120590
respectivelyFor notational simplicity we suppose both LIFS 119860 and
119861 contain only one element which stand for linguisticintuitionistic fuzzy values (LIFVs) that is the pairs 119860 =
(119904120579 119904120590) and 119861 = (119904
120583 119904])
To compare any two LIFVs the score function andaccuracy function are defined as follows
Definition 7 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
with 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] The score
function of 119860 is defined as
119878 (119860) = 119904(119905+120579minus120590)2
(5)
and the accuracy function is defined as
119867(119860) = 119904120579+120590
(6)
Thus 119860 and 119861 can be ranked by the following procedure
(1) if 119878(119860) gt 119878(119861) then 119860 gt 119861(2) if 119878(119860) = 119878(119861) and
(a) 119867(119860) = 119867(119861) then 119860 = 119861(b) 119867(119860) gt 119867(119861) then 119860 gt 119861(c) 119867(119860) lt 119867(119861) then 119860 lt 119861
It is easy to see that 0 le (119905+120579minus120590)2 le 119905 and 0 le 120579+120590 le 119905which means 119904
(119905+120579minus120590)2 119904120579+120590
isin 119878
Example 8 Let 119860 = (1199041 1199043) 119861 = (119904
0 1199044) and 119862 = (119904
1 1199045) be
LIFVs which are derived from 119878 = 119904120572| 1199040lt 119904120572le 1199046 120572 isin
[0 6]Applying formulas (5) and (6) we have
119878 (119860) = 1199042gt 119878 (119861) = 119878 (119862) = 119904
1
119867 (119861) = 1199044lt 119867 (119862) = 119904
6
(7)
Thus we obtain 119860 gt 119862 gt 119861
4 Aggregation Operators for LinguisticIntuitionistic Fuzzy Sets
Since the definition of LIFS is given it is necessary tointroduce the operations and computations between them
Definition 9 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
then
(1) 119860 = 119861 if and only if 119904120579= 119904120583and 119904120590= 119904]
(2) 119860119888 = (119904120590 119904120579) where 119860119888 is the complement of 119860
(3) the intersection of 119860 and 119861 119860 and 119861 =
(min(119904120579 119904120583)max(119904
120590 119904]))
(4) the union of 119860 and 119861 119860 or 119861 =
(max(119904120579 119904120583)min(119904
120590 119904]))
Motivated by 119905-norm and 119905-conorm we propose thefollowing operation laws for linguistic variables
Definition 10 Considering any two linguistic terms 119904120572 119904120573isin
119878 = 119904120574| 1199040lt 119904120574le 119904119905 120574 isin [0 119905] the add and multiply
operations of linguistic variable are defined as follows
119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905119878(120572119905120573119905)
(8)
119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905119879(120572119905120573119905)
(9)
where 119878(120572119905 120573119905) and 119879(120572119905 120573119905) are 119905-conorm and 119905-normrespectively
Since 119878(120572119905 120573119905) 119879(120572119905 120573119905) isin [0 1] we have 119905119878(120572119905
120573119905) 119905119879(120572119905 120573119905) isin [0 119905] which indicate the operationresults match the original linguistic term set 119878 that is119904119905119878(120572119905120573119905)
119904119905119879(120572119905120573119905)
isin 119878 In addition it is worth noting thatbecause of the monotonicity of 119905-conorm and 119905-norm thevalues of function 119905119878(120572119905 120573119905) and 119905119879(120572119905 120573119905) aremonoton-ically increasing with the increasing of 120572 and 120573 which meansthe operation results obtained by (8) and (9) are in accordwith our intuition
If we take the well-known 119878119875(120572119905 120573119905) and 119879
119875(120572119905 120573119905)
into (8) and (9) respectively then they can be rewritten asfollows
119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905(120572119905+120573119905minus120572120573119905
2)= 119904120572+120573minus120572120573119905
119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905(120572120573119905
2)= 119904120572120573119905
(10)
Example 11 Let 119878 = 119904120572| 1199040lt 119904120572le 1199046 120572 isin [0 6] Applying
(10) we have 1199043oplus 1199044= 1199043+4minus3times46
= 1199045 1199044oplus 1199045= 1199044+5minus4times56
=
1199045667
1199043otimes 1199044= 1199043times46
= 1199042 and 119904
4otimes 1199045= 1199044times56
= 1199043333
Thus we obtain 119904
3oplus 1199044lt 1199044oplus 1199045and 1199043otimes 1199044lt 1199044otimes 1199045 Such
results seem to be intuitive and can be easily accepted
Alternatively if we take the operation laws ofDefinition 3we have 119904
3oplus 1199044= 1199047notin 119878 119904
4otimes 1199045= 11990420
notin 119878 where thesubscripts of 119904
7and 119904
20are bigger than the cardinality of
linguistic term set 119878 In addition if we extend the discreteterm set 119878 to a continuous term set 119878
1015840
= 119904120572
| 120572 isin
[0 119902] [35] where 119902 (119902 gt 119905) is a sufficiently large positive
4 Journal of Applied Mathematics
integer there is an unavoidable question on how to definethe semantics for 119904
7and 11990420 Obviously 119904
7or 11990420has different
semantics in different linguistic term set 119878 with differentcardinalities As a result it is unrealistic to assign semanticsto a given linguistic value 119904
120572derived from linguistic term set
with variable cardinality If we follow the method of 119904120572oplus 119904120573=
min119904120572+120573
119904119905 and 119904
120572otimes 119904120573= min119904
120572120573 119904119905 [36] for 119904
120572119904120573isin 119878 =
119904120574
| 1199040
lt 119904120574
le 119904119905 120574 isin [0 119905] then we have 119904
3oplus 1199044
=
1199044oplus 1199045= 1199046and 1199043otimes 1199044= 1199044otimes 1199045= 1199046 Such results seem to be
counter-intuitive and may not be easily accepted Applying(10) we can overcome the limitations that the subscripts ofthe linguistic variable are bigger than the cardinality of thecorresponding linguistic term set 119878 and obtain results agreedwith our intuition
Based on (10) we can get the following operation laws forLIFVs
Definition 12 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with
120582 gt 0 then
119860 oplus 119861 = (119904119905(120579119905+120583119905minus120579120583119905
2) 119904119905(120590]1199052)) = (119904
120579+120583minus120579120583119905 119904120590]119905) (11)
119860 otimes 119861 = (119904119905(120579120583119905
2) 119904119905(120590119905+]119905minus120590]1199052)) = (119904
120579120583119905 119904120590+]minus120590]119905) (12)
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582 ) (13)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) (14)
Some special cases of 120582119860 and 119860120582 are obtained as follows
If 119860 = (119904120579 119904120590) = (119904119905 1199040) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905(1minus(1minus119905119905)
120582) 119904119905(0119905)
120582) = (119904119905 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905(119905119905)120582 119904119905(1minus(1minus0119905)
120582)) = (119904
119905 1199040)
(15)
If 119860 = (119904120579 119904120590) = (1199040 119904119905) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905(1minus(1minus0119905)
120582) 119904119905(119905119905)120582) = (119904
0 119904119905)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905(0119905)120582 119904119905(1minus(1minus119905119905)
120582)) = (119904
0 119904119905)
(16)
If 119860 = (119904120579 119904120590) = (1199040 1199040) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582)=(119904119905(1minus(1minus0119905)
120582) 119904119905(0119905)
120582)=(1199040 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582))=(119904119905(0119905)
120582 119904119905(1minus(1minus0119905)
120582))=(1199040 1199040)
(17)
If 120582 rarr 0 then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (1199040 119904119905)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905 1199040)
(18)
If 120582 rarr +infin then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
0 119904119905)
(19)
Theorem 13 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with
120582 1205821 1205822gt 0 Then one has
(1) 120582(119860 oplus 119861) = 120582119860 oplus 120582119861
(2) 1205821119860 oplus 120582
2119860 = (120582
1+ 1205822)119860
(3) 119860120582 otimes 119861120582= (119860 otimes 119861)
120582
(4) 1198601205821 otimes 1198601205822 = 1198601205821+1205822
Proof (1) By (11) we have 119860 oplus 119861 = (119904120579+120583minus120579120583119905
119904120590]119905) Thus
based on (13) we have
120582 (119860 oplus 119861) = (119904119905(1minus(1minus(120579+120583minus120579120583119905)119905)
120582) 119904119905((120590]119905)119905)120582)
= (119904119905(1minus(1minus120579119905)
120582(1minus120583119905)
120582) 119904119905(120590119905)
120582(]119905)120582)
(20)
Similarly since 120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) and 120582119861 =
(119904119905(1minus(1minus120583119905)
120582) 119904119905(]119905)120582) then
120582119860 oplus 120582119861
= (119904119905(1minus(1minus120579119905)
120582)+119905(1minus(1minus120583119905)
120582)minus119905(1minus(1minus120579119905)
120582)(1minus(1minus120583119905)
120582)
119904119905(120590119905)
120582119905(]119905)120582119905)
= (119904119905(1minus(1minus120579119905)
120582(1minus120583119905)
120582) 119904119905(120590119905)
120582(]119905)120582)
(21)
Hence we obtain 120582(119860 oplus 119861) = 120582119860 oplus 120582119861(2) By (13) we have 120582
1119860 = (119904
119905(1minus(1minus120579119905)1205821 ) 119904119905(120590119905)
1205821 ) and1205822119860 = (119904
119905(1minus(1minus120579119905)1205822 ) 119904119905(120590119905)
1205822 ) thus we obtain
1205821119860 oplus 120582
2119860
= (119904119905(1minus(1minus120579119905)
1205821 )+119905(1minus(1minus120579119905)1205822 )minus119905(1minus(1minus120579119905)
1205821 )(1minus(1minus120579119905)1205822 )
119904119905(120590119905)
1205821 (120590119905)1205822 )
= (119904119905(1minus120579119905)
1205821 (1minus120579119905)1205822 119904119905(120590119905)
1205821+1205822 )
= (119904119905(1minus120579119905)
1205821+1205822 119904119905(120590119905)1205821+1205822 ) = (120582
1+ 1205822) 119860
(22)
(3) By (14) we get 119860120582 = (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) and 119861
120582=
(119904119905(120583119905)
120582 119904119905(1minus(1minus]119905)120582)) thus based on (12) we have
119860120582otimes 119861120582
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582)+119905(1minus(1minus]119905)120582)minus119905(1minus(1minus120590119905)120582)(1minus(1minus]119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(23)
Journal of Applied Mathematics 5
Since 119860 otimes 119861 = (119904120579120583119905
119904120590+]minus120590]119905) then by (14) we have
(119860 otimes 119861)120582
= (119904119905(120579120583119905119905)
120582 119904119905(1minus(1minus(120590+]minus120590]119905)119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905minus]119905minus120590]1199052)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(24)
Hence we obtain 119860120582otimes 119861120582= (119860 otimes 119861)
120582(4) By (14) we have 119860
1205821 = (119904
119905(120579119905)1205821 119904119905(1minus(1minus120590119905)
1205821 )) and
1198601205822 = (119904119905(120579119905)
1205822 119904119905(1minus(1minus120590119905)1205822 )) thus we have
1198601205821otimes 1198601205822
= (119904119905(120579119905)
1205821 (120579119905)1205822
119904119905(1minus(1minus120590119905)
1205821 )+119905(1minus(1minus120590119905)1205822 )minus119905(1minus(1minus120590119905)
1205821 )(1minus(1minus120590119905)1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821 )(1minus120590119905)
1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )
) = 1198601205821+1205822
(25)
which completes the proof of Theorem 13
Motivated by the intuitionistic fuzzy aggregation oper-ators [14 15] in what follows we define some aggregationoperators for LIFVs
Definition 14 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedaveraging (LIFWA) operator is defined as
LIFWA (1198601 1198602 119860
119899) = 11990811198601oplus 11990821198602oplus sdot sdot sdot oplus 119908
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 )
(26)
where 119908 = (1199081 1199082 119908
119899)119879 is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
In Particular if 119908 = (1119899 1119899 1119899)119879 then the
LIFWA operator is reduced to a linguistic intuitionistic fuzzyaveraging (LIFA) operator that is
LIFWA (1198601 1198602 119860
119899) =
1
1198991198601oplus1
1198991198602oplus sdot sdot sdot oplus
1
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(27)
Based on Definition 14 we get some properties of theLIFWA operator
Theorem 15 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
0 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has
the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (28)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of LIFVs If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119860(119860
1 1198602 119860
119899)
(29)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(30)
Proof (1) Since 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
LIFWA (1198601 1198602 119860
119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579119905)
119908119894 ) 119904119905prod119899
119894=1(120590119905)119908119894 )
= (119904119905(1minus(1minus120579119905))
119904119905(120590119905)
) = (119904120579 119904120590)
(31)
(2) If 1199041205791015840
119894
ge 119904120579119894
that is 1205791015840119894ge 120579119894 for any 119894 then we have
1205791015840
119894ge 120579119894997904rArr 0 le 1 minus
1205791015840
119894
119905le 1 minus
120579119894
119905le 1
997904rArr (1 minus1205791015840
119894
119905)
119908119894
le (1 minus120579119894
119905)
119908119894
997904rArr
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
le
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 1 minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
ge 1 minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 119905(1minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
) ge 119905(1minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
)
(32)
Similarly when 1199041205901015840
119894
le 119904120590119894
for any 119894 we can get119905prod119899
119894=1(1205901015840
119894119905)1119899
le 119905prod119899
119894=1(120590119894119905)1119899
According to Definition 7 we obtain
(119904119905(1minusprod
119899
119894=1(1minus1205791015840
119894119905)1119899) 119904119905prod119899
119894=1(1205901015840
119894119905)1119899)
ge (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(33)
that is
LIFWA (1198601015840
1 1198601015840
2 119860
1015840
119899) ge LIFWA (119860
1 1198602 119860
119899)
(34)
6 Journal of Applied Mathematics
(3) Since min119894(119904120579119894
) le 119904120579119894
le max119894(119904120579119894
) and max119894(119904120590119894
) le
119904120590119894
le min119894(119904120590119894
) for any 119894 then based on the monotonicityof Theorem 15 we derive
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le LIFWA (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(35)
Definition 16 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted averaging (LIFOWA) operator is defined as
LIFOWA (1198601 1198602 119860
119899)
= 1199081119860(1)
oplus 1199082119860(2)
oplus sdot sdot sdot oplus 119908119899119860(119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899
119894=1(120590(119894)119905)119908119894 )
(36)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWA operator
Theorem 17 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set oflinguistic intuitionistic fuzzy values and119908 = (119908
1 1199082 119908
119899)119879
the associated weight vector of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin
[0 1] and sum119899119894=1
119908119894= 1 then one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (37)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(38)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(39)
(4) Commutativity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 0 1 2 119899)
be a set of linguistic intuitionistic fuzzy values then forany 119908
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) = 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(40)
where (1198601015840
1 1198601015840
2 119860
1015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Definition 18 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedgeometric (LIFWG) operator is defined as
LIFWG (1198601 1198602 119860
119899) = 1198601199081
1otimes 1198601199082
2otimes sdot sdot sdot otimes 119860
119908119899
119899
= (119904119905prod119899
119894=1(120579119894119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 ))
(41)
where 119908 = (1199081 1199082 119908
119899)119879is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
The LIFWG operator has the following properties
Theorem 19 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has the
following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119866 (1198601 1198602 119860
119899) = (119904
120579 119904120590) (42)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119866(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
(43)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119866 (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(44)
Definition 20 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted geometric (LIFOWG) operator is defined as
LIFOWG(1198601 1198602 119860
119899)
= 1198601199081
(1)otimes 1198601199082
(2)otimes sdot sdot sdot otimes 119860
119908119899
(119899)
= (119904119905prod119899
119894=1(120579(119894)119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590(119894)119905)119908119894 ))
(45)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWG operator
Journal of Applied Mathematics 7
Theorem 21 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119866(1198601 1198602 119860
119899) = (119904
120579 119904120590) (46)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(47)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119866(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(48)
(4) Commutativity Let1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values then for any119908
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(49)
where (1198601015840
1 1198601015840
2 1198601015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Lemma 22 (see [37 38]) Let 119909119894gt 0 120582
119894gt 0 119894 = 1 2 119899
and sum119899119894=1
120582119894= 1 then
119899
prod
119894=1
119909120582119894
119894le
119899
sum
119894=1
120582119894119909119894 (50)
with equality if and only if 1199091= 1199092= sdot sdot sdot = 119909
119899
Based on Lemma 22 we have the following theorem
Theorem 23 Let 119860119894= (119904120579119894
119904120590119894
)(119894 = 1 2 119899) be a set ofLIFVs then one has
119871119868119865119882119860(1198601 1198602 119860
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
119871119868119865119874119882119860(1198601 1198602 119860
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(51)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Proof Let 119908 = (1199081 1199082 119908
119899)119879 be the weight vector of
119860119894(119894 = 1 2 119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
by Lemma 22 we have 1 minusprod119899
119894=1(1 minus 120579
119894119905)119908119894 ge 1 minus sum
119899
119894=1119908119894(1 minus
120579119894119905) = 1minussum
119899
119894=1119908119894+sum119899
119894=1119908119894120579119894119905 = sum
119899
119894=1119908119894120579119894119905 ge prod
119899
119894=1(120579119894119905)119908119894
with equality if and only if 1205791
= 1205792
= sdot sdot sdot = 120579119899 that is
119905(1 minus prod119899
119894=1(1 minus 120579
119894119905)119908119894) ge 119905prod
119899
119894=1(120579119894119905)119908119894 with equality if and
only if 1205791= 1205792= sdot sdot sdot = 120579
119899 and prod
119899
119894=1(120590119894119905)119908119894 le sum
119899
119894=1119908119894120590119894119905 =
1 minus sum119899
119894=1119908119894(1 minus 120590
119894119905) le 1 minus prod
119899
119894=1(1 minus 120590
119894119905)119908119894 with equality if
and only if 1205901= 1205902= sdot sdot sdot = 120590
119899 that is 119905(prod119899
119894=1(120590119894119905)119908119894) le
119905(1 minus prod119899
119894=1(1 minus 120590
119894119905)119908119894) with equality if and only if 120590
1= 1205902=
sdot sdot sdot = 120590119899
Consequently by Definition 7 we obtain(119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 ) ge (119904
119905prod119899
119894=1(120579119894119905)119908119894
119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 )) with equality if and only if 120579
1= 1205792
=
sdot sdot sdot = 120579119899and 120590
1= 1205902= sdot sdot sdot = 120590
119899 that is
LIFWA (1198601 1198602 119860
119899) ge LIFWG (119860
1 1198602 119860
119899) (52)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Similarly we can also prove LIFOWA(1198601 1198602 119860
119899) ge
LIFOWG(1198601 1198602 119860
119899) with equality if and only if 119860
1=
1198602= sdot sdot sdot = 119860
119899
Besides the above properties we can derive the followingdesirable results of the LIFOWAandLIFOWGoperators
Theorem 24 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899
119894=1119908119894= 1 Then
one has the following
(1) If 119908 = (1 0 0)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= max119894
(119860119894)
(53)
(2) If 119908 = (0 0 1)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= min119894
(119860119894)
(54)
(3) If 119908119894= 1 and 119908
119895= 0 for 119895 = 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= 119860(119894)
(55)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of1198601 1198602 119860
119899
Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)
and 1198604= (1199044 1199041) be LIFVs which are derived from 119878 = 119904
120572|
1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)
119879 be theweight vector of 119860
119894(119894 = 1 2 3 4)
8 Journal of Applied Mathematics
Applying (26) and (41) we obtain the aggregated LIFVsas follows
LIFWA (1198601 1198602 119860
4) = (119904
6(1minusprod4
119894=1(1minus1205791198946)119908119894 ) 1199046prod4
119894=1(1205901198946)119908119894 )
= (1199041827
1199042781
)
LIFWG (1198601 1198602 119860
4) = (119904
6prod4
119894=1(1205791198946)119908119894 1199046(1minusprod
4
119894=1(1minus1205901198946)119908119894 ))
= (1199041516
1199043157
)
(56)
It is easy to see that
LIFWA (1198601 1198602 119860
4) = (119904
1827 1199042781
)
gt LIFWG (1198601 1198602 119860
4)
= (1199041516
1199043157
)
(57)
By Definition 7 we calculate the following values of scorefunction and accuracy function
119878 (1198601) = 119904(6+1minus2)2
= 11990425 119878 (119860
2) = 119904(6+1minus3)2
= 1199042
119878 (1198603) = 119904(6+2minus4)2
= 1199042 119878 (119860
4) = 119904(6+4minus1)2
= 11990445
119867 (1198602) = 1199041+3
= 1199044 119867 (119860
3) = 1199042+4
= 1199046
(58)
Since 119878(1198604) gt 119878(119860
1) gt 119878(119860
2) = 119878(119860
3) and 119867(119860
3) gt
119867(1198602) then
119860(1)
= 1198604= (1199044 1199041) 119860
(2)= 1198601= (1199041 1199042)
119860(3)
= 1198603= (1199042 1199044) 119860
(4)= 1198602= (1199041 1199043)
(59)
Assume that 119908 = (0155 0345 0345 0155)119879 which are
determined by the normal distribution based method [39] isthe associated weight vector of 119860
(119894) Then by (36) and (45)
we have
LIFOWA (1198601 1198602 119860
119899)
= (1199046(1minusprod
4
119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899
119894=1(120590(119894)6)119908119894)
= (1199041983
1199042430
)
LIFOWG (1198601 1198602 119860
119899)
= (1199046prod119899
119894=1(120579(119894)6)119908119894 1199046(1minusprod
119899
119894=1(1minus120590(119894)6)119908119894 ))
= (1199041575
1199042882
)
(60)
It is easy to see that
LIFOWA (1198601 1198602 119860
4) = (119904
1983 1199042430
)
gt LIFOWG (1198601 1198602 119860
4)
= (1199041575
1199042882
)
(61)
5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information
In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs
Let 119860 = (1198601 1198602 119860
119898) be the set of alternatives and
119862 = (1198621 1198622 119862
119899) the set of attributes whose weight vector
is 119908 = (1199081 1199082 119908
119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 Let
119863 = (1198631 1198632 119863
119897) be the set of decision makers Suppose
119877119896= (119903119896
119894119895)119898times119899
is the decision matrix where 119903119896
119894119895= (119904119896
120579119894119895
119904119896
120590119894119895
)
denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863
119896for alternative 119860
119894with
respect to attribute 119862119895and 119904119896
120579119894119895
119904119896
120590119894119895
isin 119878 = 119904119894| 119894 = 0 1 119905
The proposed method is described as follows
Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877
119896= (119903119896
119894119895)119898times119899
Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903
119894119895)119898times119899
119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWA (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
or 119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWG (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
(62)
where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]
Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective
overall preference values 119903119894for each alternative 119860
119894(119894 =
1 2 119898) based on the LIFWA or LIFWG operator
Step 4 Rank the alternatives in accordance with 119903119894 according
to Definition 7
6 Numerical Example
In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860
1is a car industry 119860
2
is a food company 1198603is a computer company 119860
4is
an arms industry The investment company must make adecision according to four criteria 119862
1is the risk analysis
1198622is the growth analysis 119862
3is the social-political impact
analysis 1198624is the environmental impact analysis The weight
vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =
1199040
= extremely poor 1199041
= very poor 1199042
= poor 1199043
=
slightly poor 1199044= fair 119904
5= slightly good 119904
6= good 119904
7=
very good 1199048= extremely good
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
integer there is an unavoidable question on how to definethe semantics for 119904
7and 11990420 Obviously 119904
7or 11990420has different
semantics in different linguistic term set 119878 with differentcardinalities As a result it is unrealistic to assign semanticsto a given linguistic value 119904
120572derived from linguistic term set
with variable cardinality If we follow the method of 119904120572oplus 119904120573=
min119904120572+120573
119904119905 and 119904
120572otimes 119904120573= min119904
120572120573 119904119905 [36] for 119904
120572119904120573isin 119878 =
119904120574
| 1199040
lt 119904120574
le 119904119905 120574 isin [0 119905] then we have 119904
3oplus 1199044
=
1199044oplus 1199045= 1199046and 1199043otimes 1199044= 1199044otimes 1199045= 1199046 Such results seem to be
counter-intuitive and may not be easily accepted Applying(10) we can overcome the limitations that the subscripts ofthe linguistic variable are bigger than the cardinality of thecorresponding linguistic term set 119878 and obtain results agreedwith our intuition
Based on (10) we can get the following operation laws forLIFVs
Definition 12 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with
120582 gt 0 then
119860 oplus 119861 = (119904119905(120579119905+120583119905minus120579120583119905
2) 119904119905(120590]1199052)) = (119904
120579+120583minus120579120583119905 119904120590]119905) (11)
119860 otimes 119861 = (119904119905(120579120583119905
2) 119904119905(120590119905+]119905minus120590]1199052)) = (119904
120579120583119905 119904120590+]minus120590]119905) (12)
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582 ) (13)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) (14)
Some special cases of 120582119860 and 119860120582 are obtained as follows
If 119860 = (119904120579 119904120590) = (119904119905 1199040) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905(1minus(1minus119905119905)
120582) 119904119905(0119905)
120582) = (119904119905 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905(119905119905)120582 119904119905(1minus(1minus0119905)
120582)) = (119904
119905 1199040)
(15)
If 119860 = (119904120579 119904120590) = (1199040 119904119905) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905(1minus(1minus0119905)
120582) 119904119905(119905119905)120582) = (119904
0 119904119905)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905(0119905)120582 119904119905(1minus(1minus119905119905)
120582)) = (119904
0 119904119905)
(16)
If 119860 = (119904120579 119904120590) = (1199040 1199040) then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582)=(119904119905(1minus(1minus0119905)
120582) 119904119905(0119905)
120582)=(1199040 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582))=(119904119905(0119905)
120582 119904119905(1minus(1minus0119905)
120582))=(1199040 1199040)
(17)
If 120582 rarr 0 then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (1199040 119904119905)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
119905 1199040)
(18)
If 120582 rarr +infin then
120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) = (119904119905 1199040)
119860120582= (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) = (119904
0 119904119905)
(19)
Theorem 13 Let 119860 = (119904120579 119904120590) and 119861 = (119904
120583 119904]) be two LIFVs
where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904
120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with
120582 1205821 1205822gt 0 Then one has
(1) 120582(119860 oplus 119861) = 120582119860 oplus 120582119861
(2) 1205821119860 oplus 120582
2119860 = (120582
1+ 1205822)119860
(3) 119860120582 otimes 119861120582= (119860 otimes 119861)
120582
(4) 1198601205821 otimes 1198601205822 = 1198601205821+1205822
Proof (1) By (11) we have 119860 oplus 119861 = (119904120579+120583minus120579120583119905
119904120590]119905) Thus
based on (13) we have
120582 (119860 oplus 119861) = (119904119905(1minus(1minus(120579+120583minus120579120583119905)119905)
120582) 119904119905((120590]119905)119905)120582)
= (119904119905(1minus(1minus120579119905)
120582(1minus120583119905)
120582) 119904119905(120590119905)
120582(]119905)120582)
(20)
Similarly since 120582119860 = (119904119905(1minus(1minus120579119905)
120582) 119904119905(120590119905)
120582) and 120582119861 =
(119904119905(1minus(1minus120583119905)
120582) 119904119905(]119905)120582) then
120582119860 oplus 120582119861
= (119904119905(1minus(1minus120579119905)
120582)+119905(1minus(1minus120583119905)
120582)minus119905(1minus(1minus120579119905)
120582)(1minus(1minus120583119905)
120582)
119904119905(120590119905)
120582119905(]119905)120582119905)
= (119904119905(1minus(1minus120579119905)
120582(1minus120583119905)
120582) 119904119905(120590119905)
120582(]119905)120582)
(21)
Hence we obtain 120582(119860 oplus 119861) = 120582119860 oplus 120582119861(2) By (13) we have 120582
1119860 = (119904
119905(1minus(1minus120579119905)1205821 ) 119904119905(120590119905)
1205821 ) and1205822119860 = (119904
119905(1minus(1minus120579119905)1205822 ) 119904119905(120590119905)
1205822 ) thus we obtain
1205821119860 oplus 120582
2119860
= (119904119905(1minus(1minus120579119905)
1205821 )+119905(1minus(1minus120579119905)1205822 )minus119905(1minus(1minus120579119905)
1205821 )(1minus(1minus120579119905)1205822 )
119904119905(120590119905)
1205821 (120590119905)1205822 )
= (119904119905(1minus120579119905)
1205821 (1minus120579119905)1205822 119904119905(120590119905)
1205821+1205822 )
= (119904119905(1minus120579119905)
1205821+1205822 119904119905(120590119905)1205821+1205822 ) = (120582
1+ 1205822) 119860
(22)
(3) By (14) we get 119860120582 = (119904119905(120579119905)
120582 119904119905(1minus(1minus120590119905)
120582)) and 119861
120582=
(119904119905(120583119905)
120582 119904119905(1minus(1minus]119905)120582)) thus based on (12) we have
119860120582otimes 119861120582
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582)+119905(1minus(1minus]119905)120582)minus119905(1minus(1minus120590119905)120582)(1minus(1minus]119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(23)
Journal of Applied Mathematics 5
Since 119860 otimes 119861 = (119904120579120583119905
119904120590+]minus120590]119905) then by (14) we have
(119860 otimes 119861)120582
= (119904119905(120579120583119905119905)
120582 119904119905(1minus(1minus(120590+]minus120590]119905)119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905minus]119905minus120590]1199052)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(24)
Hence we obtain 119860120582otimes 119861120582= (119860 otimes 119861)
120582(4) By (14) we have 119860
1205821 = (119904
119905(120579119905)1205821 119904119905(1minus(1minus120590119905)
1205821 )) and
1198601205822 = (119904119905(120579119905)
1205822 119904119905(1minus(1minus120590119905)1205822 )) thus we have
1198601205821otimes 1198601205822
= (119904119905(120579119905)
1205821 (120579119905)1205822
119904119905(1minus(1minus120590119905)
1205821 )+119905(1minus(1minus120590119905)1205822 )minus119905(1minus(1minus120590119905)
1205821 )(1minus(1minus120590119905)1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821 )(1minus120590119905)
1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )
) = 1198601205821+1205822
(25)
which completes the proof of Theorem 13
Motivated by the intuitionistic fuzzy aggregation oper-ators [14 15] in what follows we define some aggregationoperators for LIFVs
Definition 14 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedaveraging (LIFWA) operator is defined as
LIFWA (1198601 1198602 119860
119899) = 11990811198601oplus 11990821198602oplus sdot sdot sdot oplus 119908
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 )
(26)
where 119908 = (1199081 1199082 119908
119899)119879 is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
In Particular if 119908 = (1119899 1119899 1119899)119879 then the
LIFWA operator is reduced to a linguistic intuitionistic fuzzyaveraging (LIFA) operator that is
LIFWA (1198601 1198602 119860
119899) =
1
1198991198601oplus1
1198991198602oplus sdot sdot sdot oplus
1
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(27)
Based on Definition 14 we get some properties of theLIFWA operator
Theorem 15 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
0 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has
the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (28)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of LIFVs If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119860(119860
1 1198602 119860
119899)
(29)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(30)
Proof (1) Since 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
LIFWA (1198601 1198602 119860
119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579119905)
119908119894 ) 119904119905prod119899
119894=1(120590119905)119908119894 )
= (119904119905(1minus(1minus120579119905))
119904119905(120590119905)
) = (119904120579 119904120590)
(31)
(2) If 1199041205791015840
119894
ge 119904120579119894
that is 1205791015840119894ge 120579119894 for any 119894 then we have
1205791015840
119894ge 120579119894997904rArr 0 le 1 minus
1205791015840
119894
119905le 1 minus
120579119894
119905le 1
997904rArr (1 minus1205791015840
119894
119905)
119908119894
le (1 minus120579119894
119905)
119908119894
997904rArr
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
le
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 1 minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
ge 1 minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 119905(1minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
) ge 119905(1minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
)
(32)
Similarly when 1199041205901015840
119894
le 119904120590119894
for any 119894 we can get119905prod119899
119894=1(1205901015840
119894119905)1119899
le 119905prod119899
119894=1(120590119894119905)1119899
According to Definition 7 we obtain
(119904119905(1minusprod
119899
119894=1(1minus1205791015840
119894119905)1119899) 119904119905prod119899
119894=1(1205901015840
119894119905)1119899)
ge (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(33)
that is
LIFWA (1198601015840
1 1198601015840
2 119860
1015840
119899) ge LIFWA (119860
1 1198602 119860
119899)
(34)
6 Journal of Applied Mathematics
(3) Since min119894(119904120579119894
) le 119904120579119894
le max119894(119904120579119894
) and max119894(119904120590119894
) le
119904120590119894
le min119894(119904120590119894
) for any 119894 then based on the monotonicityof Theorem 15 we derive
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le LIFWA (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(35)
Definition 16 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted averaging (LIFOWA) operator is defined as
LIFOWA (1198601 1198602 119860
119899)
= 1199081119860(1)
oplus 1199082119860(2)
oplus sdot sdot sdot oplus 119908119899119860(119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899
119894=1(120590(119894)119905)119908119894 )
(36)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWA operator
Theorem 17 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set oflinguistic intuitionistic fuzzy values and119908 = (119908
1 1199082 119908
119899)119879
the associated weight vector of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin
[0 1] and sum119899119894=1
119908119894= 1 then one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (37)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(38)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(39)
(4) Commutativity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 0 1 2 119899)
be a set of linguistic intuitionistic fuzzy values then forany 119908
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) = 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(40)
where (1198601015840
1 1198601015840
2 119860
1015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Definition 18 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedgeometric (LIFWG) operator is defined as
LIFWG (1198601 1198602 119860
119899) = 1198601199081
1otimes 1198601199082
2otimes sdot sdot sdot otimes 119860
119908119899
119899
= (119904119905prod119899
119894=1(120579119894119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 ))
(41)
where 119908 = (1199081 1199082 119908
119899)119879is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
The LIFWG operator has the following properties
Theorem 19 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has the
following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119866 (1198601 1198602 119860
119899) = (119904
120579 119904120590) (42)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119866(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
(43)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119866 (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(44)
Definition 20 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted geometric (LIFOWG) operator is defined as
LIFOWG(1198601 1198602 119860
119899)
= 1198601199081
(1)otimes 1198601199082
(2)otimes sdot sdot sdot otimes 119860
119908119899
(119899)
= (119904119905prod119899
119894=1(120579(119894)119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590(119894)119905)119908119894 ))
(45)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWG operator
Journal of Applied Mathematics 7
Theorem 21 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119866(1198601 1198602 119860
119899) = (119904
120579 119904120590) (46)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(47)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119866(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(48)
(4) Commutativity Let1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values then for any119908
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(49)
where (1198601015840
1 1198601015840
2 1198601015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Lemma 22 (see [37 38]) Let 119909119894gt 0 120582
119894gt 0 119894 = 1 2 119899
and sum119899119894=1
120582119894= 1 then
119899
prod
119894=1
119909120582119894
119894le
119899
sum
119894=1
120582119894119909119894 (50)
with equality if and only if 1199091= 1199092= sdot sdot sdot = 119909
119899
Based on Lemma 22 we have the following theorem
Theorem 23 Let 119860119894= (119904120579119894
119904120590119894
)(119894 = 1 2 119899) be a set ofLIFVs then one has
119871119868119865119882119860(1198601 1198602 119860
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
119871119868119865119874119882119860(1198601 1198602 119860
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(51)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Proof Let 119908 = (1199081 1199082 119908
119899)119879 be the weight vector of
119860119894(119894 = 1 2 119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
by Lemma 22 we have 1 minusprod119899
119894=1(1 minus 120579
119894119905)119908119894 ge 1 minus sum
119899
119894=1119908119894(1 minus
120579119894119905) = 1minussum
119899
119894=1119908119894+sum119899
119894=1119908119894120579119894119905 = sum
119899
119894=1119908119894120579119894119905 ge prod
119899
119894=1(120579119894119905)119908119894
with equality if and only if 1205791
= 1205792
= sdot sdot sdot = 120579119899 that is
119905(1 minus prod119899
119894=1(1 minus 120579
119894119905)119908119894) ge 119905prod
119899
119894=1(120579119894119905)119908119894 with equality if and
only if 1205791= 1205792= sdot sdot sdot = 120579
119899 and prod
119899
119894=1(120590119894119905)119908119894 le sum
119899
119894=1119908119894120590119894119905 =
1 minus sum119899
119894=1119908119894(1 minus 120590
119894119905) le 1 minus prod
119899
119894=1(1 minus 120590
119894119905)119908119894 with equality if
and only if 1205901= 1205902= sdot sdot sdot = 120590
119899 that is 119905(prod119899
119894=1(120590119894119905)119908119894) le
119905(1 minus prod119899
119894=1(1 minus 120590
119894119905)119908119894) with equality if and only if 120590
1= 1205902=
sdot sdot sdot = 120590119899
Consequently by Definition 7 we obtain(119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 ) ge (119904
119905prod119899
119894=1(120579119894119905)119908119894
119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 )) with equality if and only if 120579
1= 1205792
=
sdot sdot sdot = 120579119899and 120590
1= 1205902= sdot sdot sdot = 120590
119899 that is
LIFWA (1198601 1198602 119860
119899) ge LIFWG (119860
1 1198602 119860
119899) (52)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Similarly we can also prove LIFOWA(1198601 1198602 119860
119899) ge
LIFOWG(1198601 1198602 119860
119899) with equality if and only if 119860
1=
1198602= sdot sdot sdot = 119860
119899
Besides the above properties we can derive the followingdesirable results of the LIFOWAandLIFOWGoperators
Theorem 24 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899
119894=1119908119894= 1 Then
one has the following
(1) If 119908 = (1 0 0)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= max119894
(119860119894)
(53)
(2) If 119908 = (0 0 1)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= min119894
(119860119894)
(54)
(3) If 119908119894= 1 and 119908
119895= 0 for 119895 = 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= 119860(119894)
(55)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of1198601 1198602 119860
119899
Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)
and 1198604= (1199044 1199041) be LIFVs which are derived from 119878 = 119904
120572|
1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)
119879 be theweight vector of 119860
119894(119894 = 1 2 3 4)
8 Journal of Applied Mathematics
Applying (26) and (41) we obtain the aggregated LIFVsas follows
LIFWA (1198601 1198602 119860
4) = (119904
6(1minusprod4
119894=1(1minus1205791198946)119908119894 ) 1199046prod4
119894=1(1205901198946)119908119894 )
= (1199041827
1199042781
)
LIFWG (1198601 1198602 119860
4) = (119904
6prod4
119894=1(1205791198946)119908119894 1199046(1minusprod
4
119894=1(1minus1205901198946)119908119894 ))
= (1199041516
1199043157
)
(56)
It is easy to see that
LIFWA (1198601 1198602 119860
4) = (119904
1827 1199042781
)
gt LIFWG (1198601 1198602 119860
4)
= (1199041516
1199043157
)
(57)
By Definition 7 we calculate the following values of scorefunction and accuracy function
119878 (1198601) = 119904(6+1minus2)2
= 11990425 119878 (119860
2) = 119904(6+1minus3)2
= 1199042
119878 (1198603) = 119904(6+2minus4)2
= 1199042 119878 (119860
4) = 119904(6+4minus1)2
= 11990445
119867 (1198602) = 1199041+3
= 1199044 119867 (119860
3) = 1199042+4
= 1199046
(58)
Since 119878(1198604) gt 119878(119860
1) gt 119878(119860
2) = 119878(119860
3) and 119867(119860
3) gt
119867(1198602) then
119860(1)
= 1198604= (1199044 1199041) 119860
(2)= 1198601= (1199041 1199042)
119860(3)
= 1198603= (1199042 1199044) 119860
(4)= 1198602= (1199041 1199043)
(59)
Assume that 119908 = (0155 0345 0345 0155)119879 which are
determined by the normal distribution based method [39] isthe associated weight vector of 119860
(119894) Then by (36) and (45)
we have
LIFOWA (1198601 1198602 119860
119899)
= (1199046(1minusprod
4
119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899
119894=1(120590(119894)6)119908119894)
= (1199041983
1199042430
)
LIFOWG (1198601 1198602 119860
119899)
= (1199046prod119899
119894=1(120579(119894)6)119908119894 1199046(1minusprod
119899
119894=1(1minus120590(119894)6)119908119894 ))
= (1199041575
1199042882
)
(60)
It is easy to see that
LIFOWA (1198601 1198602 119860
4) = (119904
1983 1199042430
)
gt LIFOWG (1198601 1198602 119860
4)
= (1199041575
1199042882
)
(61)
5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information
In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs
Let 119860 = (1198601 1198602 119860
119898) be the set of alternatives and
119862 = (1198621 1198622 119862
119899) the set of attributes whose weight vector
is 119908 = (1199081 1199082 119908
119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 Let
119863 = (1198631 1198632 119863
119897) be the set of decision makers Suppose
119877119896= (119903119896
119894119895)119898times119899
is the decision matrix where 119903119896
119894119895= (119904119896
120579119894119895
119904119896
120590119894119895
)
denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863
119896for alternative 119860
119894with
respect to attribute 119862119895and 119904119896
120579119894119895
119904119896
120590119894119895
isin 119878 = 119904119894| 119894 = 0 1 119905
The proposed method is described as follows
Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877
119896= (119903119896
119894119895)119898times119899
Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903
119894119895)119898times119899
119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWA (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
or 119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWG (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
(62)
where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]
Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective
overall preference values 119903119894for each alternative 119860
119894(119894 =
1 2 119898) based on the LIFWA or LIFWG operator
Step 4 Rank the alternatives in accordance with 119903119894 according
to Definition 7
6 Numerical Example
In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860
1is a car industry 119860
2
is a food company 1198603is a computer company 119860
4is
an arms industry The investment company must make adecision according to four criteria 119862
1is the risk analysis
1198622is the growth analysis 119862
3is the social-political impact
analysis 1198624is the environmental impact analysis The weight
vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =
1199040
= extremely poor 1199041
= very poor 1199042
= poor 1199043
=
slightly poor 1199044= fair 119904
5= slightly good 119904
6= good 119904
7=
very good 1199048= extremely good
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Since 119860 otimes 119861 = (119904120579120583119905
119904120590+]minus120590]119905) then by (14) we have
(119860 otimes 119861)120582
= (119904119905(120579120583119905119905)
120582 119904119905(1minus(1minus(120590+]minus120590]119905)119905)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905minus]119905minus120590]1199052)120582))
= (119904119905(120579119905)
120582(120583119905)120582 119904119905(1minus(1minus120590119905)
120582(1minus]119905)120582))
(24)
Hence we obtain 119860120582otimes 119861120582= (119860 otimes 119861)
120582(4) By (14) we have 119860
1205821 = (119904
119905(120579119905)1205821 119904119905(1minus(1minus120590119905)
1205821 )) and
1198601205822 = (119904119905(120579119905)
1205822 119904119905(1minus(1minus120590119905)1205822 )) thus we have
1198601205821otimes 1198601205822
= (119904119905(120579119905)
1205821 (120579119905)1205822
119904119905(1minus(1minus120590119905)
1205821 )+119905(1minus(1minus120590119905)1205822 )minus119905(1minus(1minus120590119905)
1205821 )(1minus(1minus120590119905)1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821 )(1minus120590119905)
1205822 ))
= (119904119905(120579119905)
1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )
) = 1198601205821+1205822
(25)
which completes the proof of Theorem 13
Motivated by the intuitionistic fuzzy aggregation oper-ators [14 15] in what follows we define some aggregationoperators for LIFVs
Definition 14 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedaveraging (LIFWA) operator is defined as
LIFWA (1198601 1198602 119860
119899) = 11990811198601oplus 11990821198602oplus sdot sdot sdot oplus 119908
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 )
(26)
where 119908 = (1199081 1199082 119908
119899)119879 is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
In Particular if 119908 = (1119899 1119899 1119899)119879 then the
LIFWA operator is reduced to a linguistic intuitionistic fuzzyaveraging (LIFA) operator that is
LIFWA (1198601 1198602 119860
119899) =
1
1198991198601oplus1
1198991198602oplus sdot sdot sdot oplus
1
119899119860119899
= (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(27)
Based on Definition 14 we get some properties of theLIFWA operator
Theorem 15 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
0 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has
the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (28)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of LIFVs If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119860(119860
1 1198602 119860
119899)
(29)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(30)
Proof (1) Since 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
LIFWA (1198601 1198602 119860
119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579119905)
119908119894 ) 119904119905prod119899
119894=1(120590119905)119908119894 )
= (119904119905(1minus(1minus120579119905))
119904119905(120590119905)
) = (119904120579 119904120590)
(31)
(2) If 1199041205791015840
119894
ge 119904120579119894
that is 1205791015840119894ge 120579119894 for any 119894 then we have
1205791015840
119894ge 120579119894997904rArr 0 le 1 minus
1205791015840
119894
119905le 1 minus
120579119894
119905le 1
997904rArr (1 minus1205791015840
119894
119905)
119908119894
le (1 minus120579119894
119905)
119908119894
997904rArr
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
le
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 1 minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
ge 1 minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
997904rArr 119905(1minus
119899
prod
119894=1
(1 minus1205791015840
119894
119905)
119908119894
) ge 119905(1minus
119899
prod
119894=1
(1 minus120579119894
119905)
119908119894
)
(32)
Similarly when 1199041205901015840
119894
le 119904120590119894
for any 119894 we can get119905prod119899
119894=1(1205901015840
119894119905)1119899
le 119905prod119899
119894=1(120590119894119905)1119899
According to Definition 7 we obtain
(119904119905(1minusprod
119899
119894=1(1minus1205791015840
119894119905)1119899) 119904119905prod119899
119894=1(1205901015840
119894119905)1119899)
ge (119904119905(1minusprod
119899
119894=1(1minus120579119894119905)1119899) 119904119905prod119899
119894=1(120590119894119905)1119899)
(33)
that is
LIFWA (1198601015840
1 1198601015840
2 119860
1015840
119899) ge LIFWA (119860
1 1198602 119860
119899)
(34)
6 Journal of Applied Mathematics
(3) Since min119894(119904120579119894
) le 119904120579119894
le max119894(119904120579119894
) and max119894(119904120590119894
) le
119904120590119894
le min119894(119904120590119894
) for any 119894 then based on the monotonicityof Theorem 15 we derive
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le LIFWA (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(35)
Definition 16 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted averaging (LIFOWA) operator is defined as
LIFOWA (1198601 1198602 119860
119899)
= 1199081119860(1)
oplus 1199082119860(2)
oplus sdot sdot sdot oplus 119908119899119860(119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899
119894=1(120590(119894)119905)119908119894 )
(36)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWA operator
Theorem 17 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set oflinguistic intuitionistic fuzzy values and119908 = (119908
1 1199082 119908
119899)119879
the associated weight vector of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin
[0 1] and sum119899119894=1
119908119894= 1 then one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (37)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(38)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(39)
(4) Commutativity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 0 1 2 119899)
be a set of linguistic intuitionistic fuzzy values then forany 119908
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) = 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(40)
where (1198601015840
1 1198601015840
2 119860
1015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Definition 18 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedgeometric (LIFWG) operator is defined as
LIFWG (1198601 1198602 119860
119899) = 1198601199081
1otimes 1198601199082
2otimes sdot sdot sdot otimes 119860
119908119899
119899
= (119904119905prod119899
119894=1(120579119894119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 ))
(41)
where 119908 = (1199081 1199082 119908
119899)119879is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
The LIFWG operator has the following properties
Theorem 19 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has the
following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119866 (1198601 1198602 119860
119899) = (119904
120579 119904120590) (42)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119866(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
(43)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119866 (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(44)
Definition 20 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted geometric (LIFOWG) operator is defined as
LIFOWG(1198601 1198602 119860
119899)
= 1198601199081
(1)otimes 1198601199082
(2)otimes sdot sdot sdot otimes 119860
119908119899
(119899)
= (119904119905prod119899
119894=1(120579(119894)119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590(119894)119905)119908119894 ))
(45)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWG operator
Journal of Applied Mathematics 7
Theorem 21 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119866(1198601 1198602 119860
119899) = (119904
120579 119904120590) (46)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(47)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119866(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(48)
(4) Commutativity Let1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values then for any119908
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(49)
where (1198601015840
1 1198601015840
2 1198601015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Lemma 22 (see [37 38]) Let 119909119894gt 0 120582
119894gt 0 119894 = 1 2 119899
and sum119899119894=1
120582119894= 1 then
119899
prod
119894=1
119909120582119894
119894le
119899
sum
119894=1
120582119894119909119894 (50)
with equality if and only if 1199091= 1199092= sdot sdot sdot = 119909
119899
Based on Lemma 22 we have the following theorem
Theorem 23 Let 119860119894= (119904120579119894
119904120590119894
)(119894 = 1 2 119899) be a set ofLIFVs then one has
119871119868119865119882119860(1198601 1198602 119860
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
119871119868119865119874119882119860(1198601 1198602 119860
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(51)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Proof Let 119908 = (1199081 1199082 119908
119899)119879 be the weight vector of
119860119894(119894 = 1 2 119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
by Lemma 22 we have 1 minusprod119899
119894=1(1 minus 120579
119894119905)119908119894 ge 1 minus sum
119899
119894=1119908119894(1 minus
120579119894119905) = 1minussum
119899
119894=1119908119894+sum119899
119894=1119908119894120579119894119905 = sum
119899
119894=1119908119894120579119894119905 ge prod
119899
119894=1(120579119894119905)119908119894
with equality if and only if 1205791
= 1205792
= sdot sdot sdot = 120579119899 that is
119905(1 minus prod119899
119894=1(1 minus 120579
119894119905)119908119894) ge 119905prod
119899
119894=1(120579119894119905)119908119894 with equality if and
only if 1205791= 1205792= sdot sdot sdot = 120579
119899 and prod
119899
119894=1(120590119894119905)119908119894 le sum
119899
119894=1119908119894120590119894119905 =
1 minus sum119899
119894=1119908119894(1 minus 120590
119894119905) le 1 minus prod
119899
119894=1(1 minus 120590
119894119905)119908119894 with equality if
and only if 1205901= 1205902= sdot sdot sdot = 120590
119899 that is 119905(prod119899
119894=1(120590119894119905)119908119894) le
119905(1 minus prod119899
119894=1(1 minus 120590
119894119905)119908119894) with equality if and only if 120590
1= 1205902=
sdot sdot sdot = 120590119899
Consequently by Definition 7 we obtain(119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 ) ge (119904
119905prod119899
119894=1(120579119894119905)119908119894
119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 )) with equality if and only if 120579
1= 1205792
=
sdot sdot sdot = 120579119899and 120590
1= 1205902= sdot sdot sdot = 120590
119899 that is
LIFWA (1198601 1198602 119860
119899) ge LIFWG (119860
1 1198602 119860
119899) (52)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Similarly we can also prove LIFOWA(1198601 1198602 119860
119899) ge
LIFOWG(1198601 1198602 119860
119899) with equality if and only if 119860
1=
1198602= sdot sdot sdot = 119860
119899
Besides the above properties we can derive the followingdesirable results of the LIFOWAandLIFOWGoperators
Theorem 24 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899
119894=1119908119894= 1 Then
one has the following
(1) If 119908 = (1 0 0)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= max119894
(119860119894)
(53)
(2) If 119908 = (0 0 1)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= min119894
(119860119894)
(54)
(3) If 119908119894= 1 and 119908
119895= 0 for 119895 = 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= 119860(119894)
(55)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of1198601 1198602 119860
119899
Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)
and 1198604= (1199044 1199041) be LIFVs which are derived from 119878 = 119904
120572|
1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)
119879 be theweight vector of 119860
119894(119894 = 1 2 3 4)
8 Journal of Applied Mathematics
Applying (26) and (41) we obtain the aggregated LIFVsas follows
LIFWA (1198601 1198602 119860
4) = (119904
6(1minusprod4
119894=1(1minus1205791198946)119908119894 ) 1199046prod4
119894=1(1205901198946)119908119894 )
= (1199041827
1199042781
)
LIFWG (1198601 1198602 119860
4) = (119904
6prod4
119894=1(1205791198946)119908119894 1199046(1minusprod
4
119894=1(1minus1205901198946)119908119894 ))
= (1199041516
1199043157
)
(56)
It is easy to see that
LIFWA (1198601 1198602 119860
4) = (119904
1827 1199042781
)
gt LIFWG (1198601 1198602 119860
4)
= (1199041516
1199043157
)
(57)
By Definition 7 we calculate the following values of scorefunction and accuracy function
119878 (1198601) = 119904(6+1minus2)2
= 11990425 119878 (119860
2) = 119904(6+1minus3)2
= 1199042
119878 (1198603) = 119904(6+2minus4)2
= 1199042 119878 (119860
4) = 119904(6+4minus1)2
= 11990445
119867 (1198602) = 1199041+3
= 1199044 119867 (119860
3) = 1199042+4
= 1199046
(58)
Since 119878(1198604) gt 119878(119860
1) gt 119878(119860
2) = 119878(119860
3) and 119867(119860
3) gt
119867(1198602) then
119860(1)
= 1198604= (1199044 1199041) 119860
(2)= 1198601= (1199041 1199042)
119860(3)
= 1198603= (1199042 1199044) 119860
(4)= 1198602= (1199041 1199043)
(59)
Assume that 119908 = (0155 0345 0345 0155)119879 which are
determined by the normal distribution based method [39] isthe associated weight vector of 119860
(119894) Then by (36) and (45)
we have
LIFOWA (1198601 1198602 119860
119899)
= (1199046(1minusprod
4
119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899
119894=1(120590(119894)6)119908119894)
= (1199041983
1199042430
)
LIFOWG (1198601 1198602 119860
119899)
= (1199046prod119899
119894=1(120579(119894)6)119908119894 1199046(1minusprod
119899
119894=1(1minus120590(119894)6)119908119894 ))
= (1199041575
1199042882
)
(60)
It is easy to see that
LIFOWA (1198601 1198602 119860
4) = (119904
1983 1199042430
)
gt LIFOWG (1198601 1198602 119860
4)
= (1199041575
1199042882
)
(61)
5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information
In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs
Let 119860 = (1198601 1198602 119860
119898) be the set of alternatives and
119862 = (1198621 1198622 119862
119899) the set of attributes whose weight vector
is 119908 = (1199081 1199082 119908
119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 Let
119863 = (1198631 1198632 119863
119897) be the set of decision makers Suppose
119877119896= (119903119896
119894119895)119898times119899
is the decision matrix where 119903119896
119894119895= (119904119896
120579119894119895
119904119896
120590119894119895
)
denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863
119896for alternative 119860
119894with
respect to attribute 119862119895and 119904119896
120579119894119895
119904119896
120590119894119895
isin 119878 = 119904119894| 119894 = 0 1 119905
The proposed method is described as follows
Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877
119896= (119903119896
119894119895)119898times119899
Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903
119894119895)119898times119899
119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWA (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
or 119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWG (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
(62)
where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]
Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective
overall preference values 119903119894for each alternative 119860
119894(119894 =
1 2 119898) based on the LIFWA or LIFWG operator
Step 4 Rank the alternatives in accordance with 119903119894 according
to Definition 7
6 Numerical Example
In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860
1is a car industry 119860
2
is a food company 1198603is a computer company 119860
4is
an arms industry The investment company must make adecision according to four criteria 119862
1is the risk analysis
1198622is the growth analysis 119862
3is the social-political impact
analysis 1198624is the environmental impact analysis The weight
vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =
1199040
= extremely poor 1199041
= very poor 1199042
= poor 1199043
=
slightly poor 1199044= fair 119904
5= slightly good 119904
6= good 119904
7=
very good 1199048= extremely good
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
(3) Since min119894(119904120579119894
) le 119904120579119894
le max119894(119904120579119894
) and max119894(119904120590119894
) le
119904120590119894
le min119894(119904120590119894
) for any 119894 then based on the monotonicityof Theorem 15 we derive
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le LIFWA (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(35)
Definition 16 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted averaging (LIFOWA) operator is defined as
LIFOWA (1198601 1198602 119860
119899)
= 1199081119860(1)
oplus 1199082119860(2)
oplus sdot sdot sdot oplus 119908119899119860(119899)
= (119904119905(1minusprod
119899
119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899
119894=1(120590(119894)119905)119908119894 )
(36)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWA operator
Theorem 17 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set oflinguistic intuitionistic fuzzy values and119908 = (119908
1 1199082 119908
119899)119879
the associated weight vector of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin
[0 1] and sum119899119894=1
119908119894= 1 then one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = (119904
120579 119904120590) (37)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(38)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119860(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(39)
(4) Commutativity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 0 1 2 119899)
be a set of linguistic intuitionistic fuzzy values then forany 119908
119871119868119865119874119882119860(1198601015840
1 1198601015840
2 119860
1015840
119899) = 119871119868119865119874119882119860(119860
1 1198602 119860
119899)
(40)
where (1198601015840
1 1198601015840
2 119860
1015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Definition 18 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedgeometric (LIFWG) operator is defined as
LIFWG (1198601 1198602 119860
119899) = 1198601199081
1otimes 1198601199082
2otimes sdot sdot sdot otimes 119860
119908119899
119899
= (119904119905prod119899
119894=1(120579119894119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 ))
(41)
where 119908 = (1199081 1199082 119908
119899)119879is the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
The LIFWG operator has the following properties
Theorem 19 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the weight vector of 119860
119894(119894 =
1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then one has the
following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119882119866 (1198601 1198602 119860
119899) = (119904
120579 119904120590) (42)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119882119866(1198601015840
1 1198601015840
2 119860
1015840
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
(43)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119882119866 (1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(44)
Definition 20 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted geometric (LIFOWG) operator is defined as
LIFOWG(1198601 1198602 119860
119899)
= 1198601199081
(1)otimes 1198601199082
(2)otimes sdot sdot sdot otimes 119860
119908119899
(119899)
= (119904119905prod119899
119894=1(120579(119894)119905)119908119894 119904119905(1minusprod
119899
119894=1(1minus120590(119894)119905)119908119894 ))
(45)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of 1198601 1198602 119860
119899
and 119908 = (1199081 1199082 119908
119899)119879 is the associated weight vector of
119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1
Similar to Theorem 15 we have some properties of theLIFOWG operator
Journal of Applied Mathematics 7
Theorem 21 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119866(1198601 1198602 119860
119899) = (119904
120579 119904120590) (46)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(47)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119866(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(48)
(4) Commutativity Let1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values then for any119908
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(49)
where (1198601015840
1 1198601015840
2 1198601015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Lemma 22 (see [37 38]) Let 119909119894gt 0 120582
119894gt 0 119894 = 1 2 119899
and sum119899119894=1
120582119894= 1 then
119899
prod
119894=1
119909120582119894
119894le
119899
sum
119894=1
120582119894119909119894 (50)
with equality if and only if 1199091= 1199092= sdot sdot sdot = 119909
119899
Based on Lemma 22 we have the following theorem
Theorem 23 Let 119860119894= (119904120579119894
119904120590119894
)(119894 = 1 2 119899) be a set ofLIFVs then one has
119871119868119865119882119860(1198601 1198602 119860
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
119871119868119865119874119882119860(1198601 1198602 119860
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(51)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Proof Let 119908 = (1199081 1199082 119908
119899)119879 be the weight vector of
119860119894(119894 = 1 2 119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
by Lemma 22 we have 1 minusprod119899
119894=1(1 minus 120579
119894119905)119908119894 ge 1 minus sum
119899
119894=1119908119894(1 minus
120579119894119905) = 1minussum
119899
119894=1119908119894+sum119899
119894=1119908119894120579119894119905 = sum
119899
119894=1119908119894120579119894119905 ge prod
119899
119894=1(120579119894119905)119908119894
with equality if and only if 1205791
= 1205792
= sdot sdot sdot = 120579119899 that is
119905(1 minus prod119899
119894=1(1 minus 120579
119894119905)119908119894) ge 119905prod
119899
119894=1(120579119894119905)119908119894 with equality if and
only if 1205791= 1205792= sdot sdot sdot = 120579
119899 and prod
119899
119894=1(120590119894119905)119908119894 le sum
119899
119894=1119908119894120590119894119905 =
1 minus sum119899
119894=1119908119894(1 minus 120590
119894119905) le 1 minus prod
119899
119894=1(1 minus 120590
119894119905)119908119894 with equality if
and only if 1205901= 1205902= sdot sdot sdot = 120590
119899 that is 119905(prod119899
119894=1(120590119894119905)119908119894) le
119905(1 minus prod119899
119894=1(1 minus 120590
119894119905)119908119894) with equality if and only if 120590
1= 1205902=
sdot sdot sdot = 120590119899
Consequently by Definition 7 we obtain(119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 ) ge (119904
119905prod119899
119894=1(120579119894119905)119908119894
119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 )) with equality if and only if 120579
1= 1205792
=
sdot sdot sdot = 120579119899and 120590
1= 1205902= sdot sdot sdot = 120590
119899 that is
LIFWA (1198601 1198602 119860
119899) ge LIFWG (119860
1 1198602 119860
119899) (52)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Similarly we can also prove LIFOWA(1198601 1198602 119860
119899) ge
LIFOWG(1198601 1198602 119860
119899) with equality if and only if 119860
1=
1198602= sdot sdot sdot = 119860
119899
Besides the above properties we can derive the followingdesirable results of the LIFOWAandLIFOWGoperators
Theorem 24 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899
119894=1119908119894= 1 Then
one has the following
(1) If 119908 = (1 0 0)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= max119894
(119860119894)
(53)
(2) If 119908 = (0 0 1)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= min119894
(119860119894)
(54)
(3) If 119908119894= 1 and 119908
119895= 0 for 119895 = 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= 119860(119894)
(55)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of1198601 1198602 119860
119899
Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)
and 1198604= (1199044 1199041) be LIFVs which are derived from 119878 = 119904
120572|
1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)
119879 be theweight vector of 119860
119894(119894 = 1 2 3 4)
8 Journal of Applied Mathematics
Applying (26) and (41) we obtain the aggregated LIFVsas follows
LIFWA (1198601 1198602 119860
4) = (119904
6(1minusprod4
119894=1(1minus1205791198946)119908119894 ) 1199046prod4
119894=1(1205901198946)119908119894 )
= (1199041827
1199042781
)
LIFWG (1198601 1198602 119860
4) = (119904
6prod4
119894=1(1205791198946)119908119894 1199046(1minusprod
4
119894=1(1minus1205901198946)119908119894 ))
= (1199041516
1199043157
)
(56)
It is easy to see that
LIFWA (1198601 1198602 119860
4) = (119904
1827 1199042781
)
gt LIFWG (1198601 1198602 119860
4)
= (1199041516
1199043157
)
(57)
By Definition 7 we calculate the following values of scorefunction and accuracy function
119878 (1198601) = 119904(6+1minus2)2
= 11990425 119878 (119860
2) = 119904(6+1minus3)2
= 1199042
119878 (1198603) = 119904(6+2minus4)2
= 1199042 119878 (119860
4) = 119904(6+4minus1)2
= 11990445
119867 (1198602) = 1199041+3
= 1199044 119867 (119860
3) = 1199042+4
= 1199046
(58)
Since 119878(1198604) gt 119878(119860
1) gt 119878(119860
2) = 119878(119860
3) and 119867(119860
3) gt
119867(1198602) then
119860(1)
= 1198604= (1199044 1199041) 119860
(2)= 1198601= (1199041 1199042)
119860(3)
= 1198603= (1199042 1199044) 119860
(4)= 1198602= (1199041 1199043)
(59)
Assume that 119908 = (0155 0345 0345 0155)119879 which are
determined by the normal distribution based method [39] isthe associated weight vector of 119860
(119894) Then by (36) and (45)
we have
LIFOWA (1198601 1198602 119860
119899)
= (1199046(1minusprod
4
119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899
119894=1(120590(119894)6)119908119894)
= (1199041983
1199042430
)
LIFOWG (1198601 1198602 119860
119899)
= (1199046prod119899
119894=1(120579(119894)6)119908119894 1199046(1minusprod
119899
119894=1(1minus120590(119894)6)119908119894 ))
= (1199041575
1199042882
)
(60)
It is easy to see that
LIFOWA (1198601 1198602 119860
4) = (119904
1983 1199042430
)
gt LIFOWG (1198601 1198602 119860
4)
= (1199041575
1199042882
)
(61)
5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information
In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs
Let 119860 = (1198601 1198602 119860
119898) be the set of alternatives and
119862 = (1198621 1198622 119862
119899) the set of attributes whose weight vector
is 119908 = (1199081 1199082 119908
119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 Let
119863 = (1198631 1198632 119863
119897) be the set of decision makers Suppose
119877119896= (119903119896
119894119895)119898times119899
is the decision matrix where 119903119896
119894119895= (119904119896
120579119894119895
119904119896
120590119894119895
)
denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863
119896for alternative 119860
119894with
respect to attribute 119862119895and 119904119896
120579119894119895
119904119896
120590119894119895
isin 119878 = 119904119894| 119894 = 0 1 119905
The proposed method is described as follows
Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877
119896= (119903119896
119894119895)119898times119899
Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903
119894119895)119898times119899
119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWA (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
or 119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWG (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
(62)
where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]
Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective
overall preference values 119903119894for each alternative 119860
119894(119894 =
1 2 119898) based on the LIFWA or LIFWG operator
Step 4 Rank the alternatives in accordance with 119903119894 according
to Definition 7
6 Numerical Example
In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860
1is a car industry 119860
2
is a food company 1198603is a computer company 119860
4is
an arms industry The investment company must make adecision according to four criteria 119862
1is the risk analysis
1198622is the growth analysis 119862
3is the social-political impact
analysis 1198624is the environmental impact analysis The weight
vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =
1199040
= extremely poor 1199041
= very poor 1199042
= poor 1199043
=
slightly poor 1199044= fair 119904
5= slightly good 119904
6= good 119904
7=
very good 1199048= extremely good
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
Theorem 21 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908
119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
one has the following
(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that
is 119860119894= (119904120579119894
119904120590119894
) = (119904120579 119904120590) for any 119894 then
119871119868119865119874119882119866(1198601 1198602 119860
119899) = (119904
120579 119904120590) (46)
(2) Monotonicity Let 1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904
1205791015840
119894
ge 119904120579119894
and 1199041205901015840
119894
le 119904120590119894
for any 119894 then
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(47)
for any 119908(3) Boundary Consider
(min119894
(119904120579119894
) max119894
(119904120590119894
)) le 119871119868119865119874119882119866(1198601 1198602 119860
119899)
le (max119894
(119904120579119894
) min119894
(119904120590119894
))
(48)
(4) Commutativity Let1198601015840119894= (1199041205791015840
119894
1199041205901015840
119894
) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values then for any119908
119871119868119865119874119882119866(1198601015840
1 1198601015840
2 1198601015840
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(49)
where (1198601015840
1 1198601015840
2 1198601015840
119899) is any permutation of
(1198601 1198602 119860
119899)
Lemma 22 (see [37 38]) Let 119909119894gt 0 120582
119894gt 0 119894 = 1 2 119899
and sum119899119894=1
120582119894= 1 then
119899
prod
119894=1
119909120582119894
119894le
119899
sum
119894=1
120582119894119909119894 (50)
with equality if and only if 1199091= 1199092= sdot sdot sdot = 119909
119899
Based on Lemma 22 we have the following theorem
Theorem 23 Let 119860119894= (119904120579119894
119904120590119894
)(119894 = 1 2 119899) be a set ofLIFVs then one has
119871119868119865119882119860(1198601 1198602 119860
119899) ge 119871119868119865119882119866 (119860
1 1198602 119860
119899)
119871119868119865119874119882119860(1198601 1198602 119860
119899) ge 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
(51)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Proof Let 119908 = (1199081 1199082 119908
119899)119879 be the weight vector of
119860119894(119894 = 1 2 119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 then
by Lemma 22 we have 1 minusprod119899
119894=1(1 minus 120579
119894119905)119908119894 ge 1 minus sum
119899
119894=1119908119894(1 minus
120579119894119905) = 1minussum
119899
119894=1119908119894+sum119899
119894=1119908119894120579119894119905 = sum
119899
119894=1119908119894120579119894119905 ge prod
119899
119894=1(120579119894119905)119908119894
with equality if and only if 1205791
= 1205792
= sdot sdot sdot = 120579119899 that is
119905(1 minus prod119899
119894=1(1 minus 120579
119894119905)119908119894) ge 119905prod
119899
119894=1(120579119894119905)119908119894 with equality if and
only if 1205791= 1205792= sdot sdot sdot = 120579
119899 and prod
119899
119894=1(120590119894119905)119908119894 le sum
119899
119894=1119908119894120590119894119905 =
1 minus sum119899
119894=1119908119894(1 minus 120590
119894119905) le 1 minus prod
119899
119894=1(1 minus 120590
119894119905)119908119894 with equality if
and only if 1205901= 1205902= sdot sdot sdot = 120590
119899 that is 119905(prod119899
119894=1(120590119894119905)119908119894) le
119905(1 minus prod119899
119894=1(1 minus 120590
119894119905)119908119894) with equality if and only if 120590
1= 1205902=
sdot sdot sdot = 120590119899
Consequently by Definition 7 we obtain(119904119905(1minusprod
119899
119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899
119894=1(120590119894119905)119908119894 ) ge (119904
119905prod119899
119894=1(120579119894119905)119908119894
119904119905(1minusprod
119899
119894=1(1minus120590119894119905)119908119894 )) with equality if and only if 120579
1= 1205792
=
sdot sdot sdot = 120579119899and 120590
1= 1205902= sdot sdot sdot = 120590
119899 that is
LIFWA (1198601 1198602 119860
119899) ge LIFWG (119860
1 1198602 119860
119899) (52)
with equality if and only if 1198601= 1198602= sdot sdot sdot = 119860
119899
Similarly we can also prove LIFOWA(1198601 1198602 119860
119899) ge
LIFOWG(1198601 1198602 119860
119899) with equality if and only if 119860
1=
1198602= sdot sdot sdot = 119860
119899
Besides the above properties we can derive the followingdesirable results of the LIFOWAandLIFOWGoperators
Theorem 24 Let 119860119894= (119904120579119894
119904120590119894
) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908
1 1199082 119908119899)119879 the associated weight vector
of 119860(119894)
(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899
119894=1119908119894= 1 Then
one has the following
(1) If 119908 = (1 0 0)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= max119894
(119860119894)
(53)
(2) If 119908 = (0 0 1)119879 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= min119894
(119860119894)
(54)
(3) If 119908119894= 1 and 119908
119895= 0 for 119895 = 119894 then
119871119868119865119874119882119860(1198601 1198602 119860
119899) = 119871119868119865119874119882119866(119860
1 1198602 119860
119899)
= 119860(119894)
(55)
where 119860(119894)
= (119904120579(119894)
119904120590(119894)
) is the 119894th largest of1198601 1198602 119860
119899
Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)
and 1198604= (1199044 1199041) be LIFVs which are derived from 119878 = 119904
120572|
1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)
119879 be theweight vector of 119860
119894(119894 = 1 2 3 4)
8 Journal of Applied Mathematics
Applying (26) and (41) we obtain the aggregated LIFVsas follows
LIFWA (1198601 1198602 119860
4) = (119904
6(1minusprod4
119894=1(1minus1205791198946)119908119894 ) 1199046prod4
119894=1(1205901198946)119908119894 )
= (1199041827
1199042781
)
LIFWG (1198601 1198602 119860
4) = (119904
6prod4
119894=1(1205791198946)119908119894 1199046(1minusprod
4
119894=1(1minus1205901198946)119908119894 ))
= (1199041516
1199043157
)
(56)
It is easy to see that
LIFWA (1198601 1198602 119860
4) = (119904
1827 1199042781
)
gt LIFWG (1198601 1198602 119860
4)
= (1199041516
1199043157
)
(57)
By Definition 7 we calculate the following values of scorefunction and accuracy function
119878 (1198601) = 119904(6+1minus2)2
= 11990425 119878 (119860
2) = 119904(6+1minus3)2
= 1199042
119878 (1198603) = 119904(6+2minus4)2
= 1199042 119878 (119860
4) = 119904(6+4minus1)2
= 11990445
119867 (1198602) = 1199041+3
= 1199044 119867 (119860
3) = 1199042+4
= 1199046
(58)
Since 119878(1198604) gt 119878(119860
1) gt 119878(119860
2) = 119878(119860
3) and 119867(119860
3) gt
119867(1198602) then
119860(1)
= 1198604= (1199044 1199041) 119860
(2)= 1198601= (1199041 1199042)
119860(3)
= 1198603= (1199042 1199044) 119860
(4)= 1198602= (1199041 1199043)
(59)
Assume that 119908 = (0155 0345 0345 0155)119879 which are
determined by the normal distribution based method [39] isthe associated weight vector of 119860
(119894) Then by (36) and (45)
we have
LIFOWA (1198601 1198602 119860
119899)
= (1199046(1minusprod
4
119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899
119894=1(120590(119894)6)119908119894)
= (1199041983
1199042430
)
LIFOWG (1198601 1198602 119860
119899)
= (1199046prod119899
119894=1(120579(119894)6)119908119894 1199046(1minusprod
119899
119894=1(1minus120590(119894)6)119908119894 ))
= (1199041575
1199042882
)
(60)
It is easy to see that
LIFOWA (1198601 1198602 119860
4) = (119904
1983 1199042430
)
gt LIFOWG (1198601 1198602 119860
4)
= (1199041575
1199042882
)
(61)
5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information
In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs
Let 119860 = (1198601 1198602 119860
119898) be the set of alternatives and
119862 = (1198621 1198622 119862
119899) the set of attributes whose weight vector
is 119908 = (1199081 1199082 119908
119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 Let
119863 = (1198631 1198632 119863
119897) be the set of decision makers Suppose
119877119896= (119903119896
119894119895)119898times119899
is the decision matrix where 119903119896
119894119895= (119904119896
120579119894119895
119904119896
120590119894119895
)
denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863
119896for alternative 119860
119894with
respect to attribute 119862119895and 119904119896
120579119894119895
119904119896
120590119894119895
isin 119878 = 119904119894| 119894 = 0 1 119905
The proposed method is described as follows
Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877
119896= (119903119896
119894119895)119898times119899
Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903
119894119895)119898times119899
119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWA (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
or 119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWG (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
(62)
where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]
Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective
overall preference values 119903119894for each alternative 119860
119894(119894 =
1 2 119898) based on the LIFWA or LIFWG operator
Step 4 Rank the alternatives in accordance with 119903119894 according
to Definition 7
6 Numerical Example
In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860
1is a car industry 119860
2
is a food company 1198603is a computer company 119860
4is
an arms industry The investment company must make adecision according to four criteria 119862
1is the risk analysis
1198622is the growth analysis 119862
3is the social-political impact
analysis 1198624is the environmental impact analysis The weight
vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =
1199040
= extremely poor 1199041
= very poor 1199042
= poor 1199043
=
slightly poor 1199044= fair 119904
5= slightly good 119904
6= good 119904
7=
very good 1199048= extremely good
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
Applying (26) and (41) we obtain the aggregated LIFVsas follows
LIFWA (1198601 1198602 119860
4) = (119904
6(1minusprod4
119894=1(1minus1205791198946)119908119894 ) 1199046prod4
119894=1(1205901198946)119908119894 )
= (1199041827
1199042781
)
LIFWG (1198601 1198602 119860
4) = (119904
6prod4
119894=1(1205791198946)119908119894 1199046(1minusprod
4
119894=1(1minus1205901198946)119908119894 ))
= (1199041516
1199043157
)
(56)
It is easy to see that
LIFWA (1198601 1198602 119860
4) = (119904
1827 1199042781
)
gt LIFWG (1198601 1198602 119860
4)
= (1199041516
1199043157
)
(57)
By Definition 7 we calculate the following values of scorefunction and accuracy function
119878 (1198601) = 119904(6+1minus2)2
= 11990425 119878 (119860
2) = 119904(6+1minus3)2
= 1199042
119878 (1198603) = 119904(6+2minus4)2
= 1199042 119878 (119860
4) = 119904(6+4minus1)2
= 11990445
119867 (1198602) = 1199041+3
= 1199044 119867 (119860
3) = 1199042+4
= 1199046
(58)
Since 119878(1198604) gt 119878(119860
1) gt 119878(119860
2) = 119878(119860
3) and 119867(119860
3) gt
119867(1198602) then
119860(1)
= 1198604= (1199044 1199041) 119860
(2)= 1198601= (1199041 1199042)
119860(3)
= 1198603= (1199042 1199044) 119860
(4)= 1198602= (1199041 1199043)
(59)
Assume that 119908 = (0155 0345 0345 0155)119879 which are
determined by the normal distribution based method [39] isthe associated weight vector of 119860
(119894) Then by (36) and (45)
we have
LIFOWA (1198601 1198602 119860
119899)
= (1199046(1minusprod
4
119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899
119894=1(120590(119894)6)119908119894)
= (1199041983
1199042430
)
LIFOWG (1198601 1198602 119860
119899)
= (1199046prod119899
119894=1(120579(119894)6)119908119894 1199046(1minusprod
119899
119894=1(1minus120590(119894)6)119908119894 ))
= (1199041575
1199042882
)
(60)
It is easy to see that
LIFOWA (1198601 1198602 119860
4) = (119904
1983 1199042430
)
gt LIFOWG (1198601 1198602 119860
4)
= (1199041575
1199042882
)
(61)
5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information
In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs
Let 119860 = (1198601 1198602 119860
119898) be the set of alternatives and
119862 = (1198621 1198622 119862
119899) the set of attributes whose weight vector
is 119908 = (1199081 1199082 119908
119899) with 119908
119894isin [0 1] and sum
119899
119894=1119908119894= 1 Let
119863 = (1198631 1198632 119863
119897) be the set of decision makers Suppose
119877119896= (119903119896
119894119895)119898times119899
is the decision matrix where 119903119896
119894119895= (119904119896
120579119894119895
119904119896
120590119894119895
)
denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863
119896for alternative 119860
119894with
respect to attribute 119862119895and 119904119896
120579119894119895
119904119896
120590119894119895
isin 119878 = 119904119894| 119894 = 0 1 119905
The proposed method is described as follows
Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877
119896= (119903119896
119894119895)119898times119899
Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903
119894119895)119898times119899
119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWA (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
or 119903119894119895= (119904120579119894119895
119904120590119894119895
) = LIFOWG (1199031
119894119895 1199032
119894119895 119903
119897
119894119895)
(62)
where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]
Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective
overall preference values 119903119894for each alternative 119860
119894(119894 =
1 2 119898) based on the LIFWA or LIFWG operator
Step 4 Rank the alternatives in accordance with 119903119894 according
to Definition 7
6 Numerical Example
In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860
1is a car industry 119860
2
is a food company 1198603is a computer company 119860
4is
an arms industry The investment company must make adecision according to four criteria 119862
1is the risk analysis
1198622is the growth analysis 119862
3is the social-political impact
analysis 1198624is the environmental impact analysis The weight
vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =
1199040
= extremely poor 1199041
= very poor 1199042
= poor 1199043
=
slightly poor 1199044= fair 119904
5= slightly good 119904
6= good 119904
7=
very good 1199048= extremely good
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
Table 1 The decision matrix 1198771given by119863
1
1198621
1198622
1198623
1198624
1198601
(1199046 1199041) (119904
3 1199041) (119904
3 1199043) (119904
1 1199046)
1198602
(1199043 1199044) (119904
3 1199044) (119904
2 1199045) (119904
2 1199044)
1198603
(1199041 1199043) (119904
2 1199043) (119904
3 1199042) (119904
6 1199041)
1198604
(1199046 1199042) (119904
4 1199043) (119904
5 1199041) (119904
7 1199041)
Table 2 The decision matrix 1198772given by119863
2
1198621
1198622
1198623
1198624
1198601
(1199043 1199042) (119904
4 1199041) (119904
3 1199044) (119904
2 1199043)
1198602
(1199045 1199042) (119904
2 1199041) (119904
3 1199044) (119904
2 1199045)
1198603
(1199042 1199043) (119904
3 1199043) (119904
1 1199042) (119904
3 1199043)
1198604
(1199045 1199042) (119904
3 1199043) (119904
5 1199042) (119904
4 1199041)
Table 3 The decision matrix 1198773given by119863
3
1198621
1198622
1198623
1198624
1198601
(1199043 1199043) (119904
3 1199045) (119904
6 1199041) (119904
2 1199046)
1198602
(1199043 1199042) (119904
2 1199044) (119904
2 1199041) (119904
3 1199044)
1198603
(1199046 1199041) (119904
2 1199045) (119904
3 1199044) (119904
1 1199043)
1198604
(1199045 1199041) (119904
4 1199044) (119904
6 1199042) (119904
5 1199042)
Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877
119896= (119903
119896
119894119895)119898times119899
(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively
Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =
(0243 0514 0243)119879 Then utilize the LIFOWA operator
to derive the aggregated decision matrix 119877 = (119903119894119895)4times4
which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903
119894119895)119898times119899
can be derived asshown in Table 5
Step 3 Aggregate 119903119894119895
(119895 = 1 2 4) to yield the collectiveoverall preference values 119903
119894for each alternative 119860
119894(119894 =
1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6
Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7
the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860
4
In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws
Table 4 The aggregated decision matrix 119877 by LIFOWA operator
1198621
1198622
1198623
1198624
1198601
(1199043998
1199041865
) (1199043264
1199041479
) (1199043998
1199042463
) (1199041771
1199045070
)
1198602
(1199043584
1199042367
) (1199042537
1199042856
) (1199042537
1199043051
) (1199042260
1199044223
)
1198603
(1199043230
1199042297
) (1199042260
1199043397
) (1199042574
1199042856
) (1199043657
1199042297
)
1198604
(1199045282
1199041401
) (1199043777
1199043478
) (1199045282
1199041401
) (1199044720
1199041184
)
Table 5 The aggregated decision matrix 119877 by LIFOWG operator
1198621
1198622
1198623
1198624
1198601
(1199043550
1199042041
) (1199043217
1199042303
) (1199043550
1199042860
) (1199041690
1199045501
)
1198602
(1199043397
1199042563
) (1199042463
1199043417
) (1199042462
1199043727
) (1199042207
1199044270
)
1198603
(1199042207
1199042574
) (1199042207
1199043584
) (1199042719
1199043129
) (1199042719
1199042774
)
1198604
(1199045227
1199041505
) (1199043730
1199043542
) (1199045227
1199041505
) (1199044838
1199041257
)
described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908
1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and
1199084= [1199044 1199047] which can be transformed to LIFVs that is119908
1=
(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908
4= (1199044 1199041) Thus by
(11) and (12) the collective overall preference values 119903119894for each
alternative 119860119894(119894 = 1 2 4) with linguistic weights can be
obtained by the following
119903119894= (1199081otimes 1199031198941) oplus (119908
2otimes 1199031198942) oplus (119908
3otimes 1199031198943)
oplus (1199084otimes 1199031198944)
(63)
Without loss of generalization on the basis of Table 4 wehave the following results
1199031= ((1199043 1199042) otimes (1199043998
1199041865
)) oplus ((1199041 1199047) otimes (1199043264
1199041479
))
oplus ((1199042 1199044) otimes (1199043998
1199042463
)) oplus ((1199044 1199041) otimes (1199041771
1199045070
))
= (1199043200
1199041245
)
1199032= ((1199043 1199042) otimes (1199043584
1199042367
)) oplus ((1199041 1199047) otimes (1199042537
1199042856
))
oplus ((1199042 1199044) otimes (1199042537
1199043051
)) oplus ((1199044 1199041) otimes (1199042260
1199044223
))
= (1199042931
1199041221
)
1199033= ((1199043 1199042) otimes (1199043230
1199042297
)) oplus ((1199041 1199047) otimes (1199042260
1199043397
))
oplus ((1199042 1199044) otimes (1199042574
1199042856
)) oplus ((1199044 1199041) otimes (1199043657
1199042297
))
= (1199043355
1199040882
)
1199034= ((1199043 1199042) otimes (1199045282
1199041401
)) oplus ((1199041 1199047) otimes (1199043777
1199043478
))
oplus ((1199042 1199044) otimes (1199045282
1199041401
)) oplus ((1199044 1199041) otimes (1199044720
1199041184
))
= (1199045109
1199040424
)
(64)
by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
Table 6 The collective overall preference values and the rankings of alternatives
1198601
1198602
1198603
1198604
RankingLIFOWALIFWA (119904
3142 1199042874
) (1199042772
1199043199
) (1199043198
1199042495
) (1199044938
1199041435
) 1198604gt 1198603gt 1198601gt 1198602
LIFOWGLIFWG (119904
2612 1199043932
) (1199042596
1199043619
) (1199042501
1199042876
) (1199044990
1199041650
) 1198604gt 1198603gt 1198602gt 1198601
Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11
7 Conclusions
In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14
References
[1] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996
[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001
[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013
[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013
[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011
[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012
[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999
[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008
[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011
[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003
[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000
[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010
[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006
[15] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007
[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009
[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009
[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010
[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010
[20] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011
[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989
[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000
[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010
[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001
[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005
[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004
[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010
[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995
[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000
[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006
[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889
[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906
[39] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of