(c)If a k>a l,then(s k,a k)>(s l,a l).
To preserve all the given information,we extend the discrete term set S to a continuous term set S?f s l j s06s l6s q;l2?0;q g, where q P t and q is a suf?ciently large positive integer,whose ele-ments also meet all the characteristics above.If s l2S,then we call s l the original term,otherwise,we call s l the virtual term.In gen-eral,the decision maker uses the original linguistic term to evalu-ate attributes and alternatives,and the virtual linguistic terms can only appear in calculation[4,10].
2.2.Operation laws and properties for2-tuple linguistic information De?nition4.Let(s k,a k)and(s l,a l)be two2-tuples and k P0.Then the operation laws for2-tuples are de?ned as follows:
(1)(s k,a k)è(s l,a l)=D(Dà1(s k,a k)+Dà1(s l,a l));
(2)(s k,a k) (s l,a l)=D(Dà1(s k,a k)áDà1(s l,a l));
(3)k(s k,a k)=D(k Dà1(s k,a k));
(4)(s k,a k)k=D((Dà1(s k,a k))k);
(5)es k;a kTes l;a lT?DeeDà1es k;a kTTDà1es l;a lTT.
Remark 1.It should be noted that if the2-tuple linguistic information comes from different linguistic term sets(i.e.multi-granularity linguistic information),they have to be converted into the fuzzy sets de?ned in the basic linguistic term set by means of a transformation function[27],then they can be operated using the above operation laws.To avoid the operation results of De?nition4 being out of the scope[s0,s q],we can make the cardinality q+1of the extended continuous term set S large enough.
If all2-tuples(s k,a k)and(s l,a l)in De?nition4are reduced to linguistic labels s k and s l,i.e.,a k=0and a l=0,then the operation laws in De?nition4are reduced to the following operation laws:
(1)s kès l=s k+l;
(2)s k s l?s k
l
;
(3)k s k=s k k;
(4)es kTk?s
k k ;
(5)es kTs l?s
k l
.
The above are just the operation laws for linguistic labels de?ned in[4–6,32],which shows the justi?cation of De?nition4to some degree.As far as we know,however,there is less investigation on the operation laws of2-tuples.De?nition4gives the operation laws of2-tuples,which can be used for direct calculation of2-tuple linguistic information.We insist that this is an interesting and valuable work for2-tuples though it is a generalization of the operation algorithms in[4–6,32],even if it is a formal transformation.
In the following,suppose that a given linguistic term set is S={s0,s1,s2,...,s8},we give some examples to illustrate the above De?nition4.
Example 1.(s1,0.1)è(s3,0.2)=D(Dà1(s1,0.1)+Dà1(s3,0.2))= D(4.3)=(s4,0.3).Example3.2(s2,0.3)=D(2Dà1(s2,0.3))=D(2?2.3)=D(4.6)= (s5,à0.4).
Example 4.(s2,0.3)2=D((Dà1(s2,0.3))2)=D(2.32)=D(5.29)= (s5,0.29).
Example 5.es1;0:1Tes3;0:2T?DeeDà1es1;0:1TTDà1es3;0:2TT?De1:13:2T?De1:3566T?es1;0:3566T:
From De?nition4,the following Theorem1can be easily proven:
Theorem1.Let(s k,a k),(s l,a l)and(s i,a i)be three2-tuples and k, k P0.The following equalities hold:
(1)(s k,a k)è(s l,a l)=(s l,a l)è(s k,a k);
(2)(s k,a k) (s l,a l)=(s l,a l) (s k,a k);
(3)k((s k,a k)è(s l,a l))=k(s k,a k)èk(s l,a l);
(4)((s k,a k)k)k=(s k,a k)k k,(s k,a k)k (s k,a k)k=(s k,a k)k+k;
(5)[(s k,a k)è(s l,a l)] (s i,a i)=[(s k,a k) (s i,a i)]è[(s l,a l) (s i,a i)];
(6)[(s k,a k) (s l,a l)] (s i,a i)=(s k,a k) [(s l,a l) (s i,a i)].
3.Some arithmetic aggregation operators with2-tuple linguistic information
3.1.The existing2-tuple linguistic arithmetic aggregation operators
Based on De?nitions2and3,the existing arithmetic aggrega-tion operators with2-tuple linguistic information are presented in this subsection.For convenience,let T be the set of all2-tuples. De?nition5[17].Let x={(r1,a1),(r2,a2),...,(r n,a n)}be a set of2-tuples,the2-tuple arithmetic averaging TAA is de?ned as
TAAeer1;a1T;er2;a2T;...;er n;a nTT?D
1
n
X n
j?1
Dà1er j;a jT
!
:e4T
De?nition6[17].Let x={(r1,a1),(r2,a2),...,(r n,a n)}be a set of2-tuples,and w=(w1,w2,...,w n)T be the weight vector of2-tuples
(r j,a j)(j=1,2,...,n),satisfying that06w j61(j=1,2,...,n)and P n
j?1
w j?1.The2-tuple weighted average TWA is de?ned as follows:
TWA weer1;a1T;er2;a2T;...;er n;a nTT?D
X n
j?1
w j Dà1er j;a jT
!
:e5T
Especially,if w j=1/n(j=1,2,...,n),then the TWA operator is re-duced to the TAA operator.
De?nition7[17].Let x={(r1,a1),(r2,a2),...,(r n,a n)}be a set of2-tuples.The2-tuple ordered weighted average(TOWA)operator of dimension n is a mapping TOWA:T n?T so that
TOWA weer1;a1T;er2;a2T;...;er n;a nTT
?D
X n
j?1
w j Dà1er rejT;a rejTT
!
;e6T
where w=(w1,w2,...,w n)T is the weighted vector correlating with TOWA,satisfying that06w j61(j=1,2,...,n)and
P n
j?1
w j?1. (r(1),r(2),...,r(n))is a permutation of(1,2,...,n)such that(r r(jà1), a r(jà1))P(r r(j),a r(j))for any j.
S.-P.Wan/Knowledge-Based Systems45(2013)31–4033
De?nition 8[17].Let x ={(r 1,a 1),(r 2,a 2),...,(r n ,a n )}be a set of 2-tuples,and C =((c 1,b 1),(c 2,b 2),...,(c n ,b n ))T be the linguistic weighting vector of 2-tuples (r j ,a j )(j =1,2,...,n ).The extended 2-tuple weighted average (ET àWA )operator is de?ned as follows:
ET àWA C eer 1;a 1T;er 2;a 2T;...;er n ;a n TT
?D X n j ?1D à1ec j ;b j TD à1er j ;a j T
P j ?1
D ec j ;b j T !:e7T
De?nition 9[20].Let x ={(r 1,a 1),(r 2,a 2),...,(r n ,a n )}be a set of 2-tuples.The extended 2-tuple ordered weighted average (ET àOWA )
operator of dimension n is a mapping ET àOWA :T n ?T so that
ET àOWA L eer 1;a 1T;er 2;a 2T;...;er n ;a n TT
?D X n j ?1D à1el j ;g j TD à1er r ej T;a r ej TT
P j ?1
D el j ;g j T !:e8T
where L =((l 1,g 1),(l 2,g 2),...,(l n ,g n ))T is the linguistic weighted
vector correlating with ET àOWA ,(r (1),r (2),...,r (n ))is a permuta-tion of (1,2,...,n )such that (r r (j à1),a r (j à1))P (r r (j ),a r (j ))for any j .3.2.The proposed hybrid arithmetic aggregation operators with 2-tuple linguistic information
It can be seen from De?nitions 8and 9that the ET àWA oper-ator weights the 2-tuple linguistic arguments while the ET àOWA operator weights the ordered positions of the 2-tuple linguistic arguments instead of weighting the arguments themselves.There-fore,weights represent different aspects in both the ET àWA and ET àOWA operators.However,the two operators consider only one of them.To solve this drawback,based on De?nitions 2–4,some hybrid arithmetic aggregation operators with 2-tuple lin-guistic information are developed in the following.
De?nition 10.Let x ={(r 1,a 1),(r 2,a 2),...,(r n ,a n )}be a set of 2-tuples.If THWA :T n ?T so that
THWA w ;x eer 1;a 1T;er 2;a 2T;...;er n ;a n TT
?D X
n j ?1
ew j D à1er 0r ej T;a 0r ej TTT !;
e9T
where w =(w 1,w 2,...,w n )T is the weighted vector correlating with
THWA ,satisfying that 06w j 61(j =1,2,...,n )and P n
j ?1w j ?1.er 0r ej T;a 0r ej TTis the j th largest 2-tuple of 2-tuples r 0i ;a 0
i àáe1;2;...;n T
with r 0i ;a 0i àá
?n x i er i ;a i T.x =(x 1,x 2,...,x n )T is the weighting vec-tor of 2-tuples (r j ,a j )(j =1,2,...,n ),satisfying that 06x j -61(j =1,2,...,n )and P n
j ?1x j ?1.n is the balancing coef?cient (in this case,if x =(x 1,x 2,...,x n )T becomes ((1/n ,1/n ,...,1/n )),
then r 0i ;a 0i àá
becomes (r i ,a i )(1,2,...,n ).Then the function THWA is called the 2-tuple hybrid weighted arithmetic average operator of dimension n .
Example 6.Assume
that,(r 1,a 1)=(s 1,0.1),(r 2,a 2)=(s 3,0.3),
(r 3,a 3)=(s 2,0.2),(r 4,a 4)=(s 4,0.3),w =(0.2,0.3,0.3,0.2)T
and x =(0.1,0.4,
0.3,0.2)T ,then,r 01;a 01àá?4?0:1es 1;0:1T?es 0;0:44T;r 02;a 02àá
?1:6es 3;0:3T
?es 5;0:28T;r 03;a 03àá?1:2es 2;0:2T?es 3;à0:36T;r 04;a 04àá
?0:8es 4;0:3T?
es 3;0:44T,therefore,r 0r e1T;a 0r e1T ?es 5;0:28T;r 0r e2T;a 0
r e2T ?es 3;0:44T;r 0r e3T;a 0r e3T ?es 3;à0:36Tand r 0r e4T;a 0
r e4T ?es 0;0:44T.Thus,
THWA w ;x eer 1;a 1T;er 2;a 2T;er 3;a 3T;er 4;a 4TT
?D X
4j ?1
w j D à1r 0r ej T;a 0r ej T
!?D e2:968T?es 3;à0:032T:Theorem 2.The TOWA operator is a special case of the THWA operator.
Proof.Let x j =1/n (j =1,2,...,n ),then r 0i ;a 0i àá
?n x i er i ;a i T?er i ;a i Te1;2;...;n T.This completes the proof of Theorem 2.h
Theorem 3.The TWA operator is a special case of the THWA operator.Proof.Let w j =1/n (j =1,2,...,n ),then
THWA w ;x eer 1;a 1T;er 2;a 2T;...;er n ;a n TT?D X n j ?1
1n D à1er 0
r ej T;a 0r ej TT
!?D X n j ?11D à1en x j er j ;a j TT !?D
X n j ?1
1
n x j D à1eer j ;a j TT !?D
X
n j ?1
ex j D à1er j ;a j TT !?TWA x eer 1;a 1T;er 2;a 2T;...;er n ;a n TT;which completes the proof of Theorem 3.h
From Theorems 2and 3,we know that,the THWA operator ?rst
weights the given arguments,then reorders the weighted argu-ments in descending order and weights these ordered arguments,and ?nally aggregates all the weighted arguments into a collective one.The THWA operator generalizes both the TWA operator and the TOWA operator.The THWA operator re?ects the important de-grees of both the given 2-tuples and the ordered positions of the 2-tuples.
De?nition 11.Let x ={(r 1,a 1),(r 2,a 2),...,(r n ,a n )}be a set of 2-tuples.If T àHLWA :T n ?T so that
T àHLWA L ;x eer 1;a 1T;er 2;a 2T;...;er n ;a n TT
?D X n j ?1D à1el j ;g j T
P j ?1D el j ;j T
D à1er 0r ej T;a 0r ej TT ! !;e10T
where L =((l 1,g 1),(l 2,g 2),...,(l n ,g n ))T is the 2-tuple linguistic weighted vector correlating with T àHLWA .er 0r ej T;a 0r ej TTis the j th largest 2-tuple of 2-tuples er 0i ;a 0i Tei ?1;2;...;n Twith er 0i ;a 0i T?n x i er i ;a i T;x ?ex 1;x 2;...;x n TT is the weighting vector of 2-tuples (r j ,a j )(j =1,2,...,n ),satisfying that 06x j 61(j =1,2,...,n )and P n
j ?1x j ?1.n is the balancing coef?cient (in this case,if x =(x 1,-x 2,...,x n )T
becomes (1/n ,1/n ,...,1/n )),then r 0i ;a 0i àágoes to (r i ,a i )(-i =1,2,...,n ).Then the function T àHLWA is called the 2-tuple hybrid linguistic weighted arithmetic average operator of dimen-sion n .
Example 7.Assume
that (l 1,g 1)=(s 3,0.4),(l 2,g 2)=(s 2,0.2),
(l 3,g 3)=(s 1,0.1),(l 4,g 4)=(s 5,0.2),(r 1,a 1)=(s 1,0.1),(r 2,a 2)=(s 3,0.3),(r 3,a 3)=(s 2,0.2),(r 4,a 4)=(s 4,0.3)and x =(0.1,0.4,0.3,0.2)T ,then,r 01;a 01àá?4?0:1es 1;0:1T?es 0;0:44T;r 02;a 02àá
?1:6es 3;0:3T?es 5;0:28T;r 03;a 03àá?1:2es 2;0:2T?es 3;à0:36Tand r 04;a 04àá?0:8es 4;0:3T?es 3;0:44T.Therefore,r 0r e1T; a 0r e1TT?es 5;0:28T;r 0r e2T;a 0r e2T ?es 3;0:44T;r 0r e3T;a 0
r e3T
?es 3;à0:36Tand r 0r e4T;a 0r e4T ?es 0;0:44T.Thus,
T àHLWA L ;x eer 1;a 1T;er 2;a 2T;er 3;a 3T;er 4;a 4TT
?D X 4j ?1D à1el j ;g j T
P j ?1
D el j ;j TD à1r 0r ej T;a 0r ej T ! !?D e5:5808T?es 6;à0:4192T:
Theorem 4.The ET àOWA operator is a special case of the T àHLWA
operator.
34S.-P.Wan /Knowledge-Based Systems 45(2013)31–40
Proof.Let x j =1/n (j =1,2,...,n ),then r 0i ;a 0i àá
?n x i er i ;a i T?er i ;a i T;ei ?1;2;...;n T.This completes the proof of Theorem 4.h De?nition 12.Let x ={(r 1,a 1),(r 2,a 2),...,(r n ,a n )}be a set of 2-tuples.If ET àHLWA :T n ?T so that
ET àHLWA L ;C eer 1;a 1T;er 2;a 2T;...;er n ;a n TT
?D X n j ?1D à1el j ;g j TX n j ?1
D à1el j ;g j T
D à1r 0r ej T;a 0r ej T 0B B B B @1C C C C A 0B B B B @1C C C C A
;e11T
where L =((l 1,g 1),(l 2,g 2),...,(l n ,g n ))T is the 2-tuple linguistic
weighted vector correlating with ET àHLWA .r 0r ej T;a 0
r ej T is the j th largest 2-tuple of 2-tuples er 0i ;a 0i Tei ?1;2;...;n Twith er 0i ;a 0i T?n ec i ;b i T er i ;a i T;C ?eec 1;b 1T;ec 2;b 2T;...;ec n ;b n TTT is the 2-tuple lin-guistic weighting vector of 2-tuples (r j ,a j ),n is the balancing coef?-cient.Then the function ET àHLWA is called the extended 2-tuple hybrid linguistic weighted arithmetic average operator of dimen-sion n .
Example 8.Assume that (l 1,g 1)=(s 3,0.4),(l 2,g 2)=(s 2,0.2),(l 3,g 3)=(s 1,0.1),(l 4,g 4)=(s 5,0.2),(r 1,a 1)=(s 3,0.1),(r 2,a 2)=(s 3,0.3),(r 3,a 3)=(s 1,0.2),(r 4,a 4)=(s 2,0.3),(c 1,b 1)=(s 1,à0.4),(c 2,b 2)=(s 0,0.1),(c 3,b 3)=(s 1,0.2)and (c 4,b 4)=(s 1,à0.3),then,er 01;a 01T?4es 1;à0:4T es 3;0:1T?es 3;à0:36T;er 02;a 02T?4es 0;0:1T es 3;0:3T?es 1;0:32T;er 03;a 03T?4es 1;0:2T es 1;0:2T?es 6;à0:24Tand er 04;a 04T?4es 1;à0:3T es 2;0:3T?es 6;0:44T.Therefore,er 0r e1T;a 0r e1TT?es 6;0:44T;er 0r e2T;a 0r e2TT?es 6;à0:24T;er 0r e3T;a 0r e3TT?es 3;à0:36Tand er 0r e4T;a 0r e4TT?es 1;0:32T.Thus,
ET àHLWA L ;C eer 1;a 1T;er 2;a 2T;er 3;a 3T;er 4;a 4TT?D X 4j ?1D à1el j ;g j TP 4j ?1D à1
el j ;g j TD à1er 0r ej T;a 0
r ej TT ! !?D e14:4712T?es 4;à0:2743T:
Remark 2.If the correlated 2-tuple linguistic weighted vector L =((l 1,g 1),(l 2,g 2),...,(l n ,g n ))T in De?nitions 11and 12is reduced to linguistic weighted vector L =(l 1,l 2,...,l n )T ,we can converted L =(l 1,l 2,...,l n )T to 2-tuple linguistic weighted vector L =((l 1,g 1),(l 2,g 2),...,(l n ,g n ))T by using Eq.(3),then De?nitions 11and 12are still validated.
Remark 3.Xu [33]proposed the linguistic hybrid arithmetic aver-aging operator (i.e.,LHAA)for virtual linguistic variables.In the LHAA operator,the weight vector x =(x 1,x 2,...,x n )T of the vir-tual linguistic variables and the associated weight vector w =(w 1,-w 2,...,w n )T are all just in the form of real numbers rather than linguistic variables or 2-tuples.Whereas,for the proposed T-HLWA operator in this paper,the associated weight vector L =((l 1,g 1),(l 2-,g 2),...,(l n ,g n ))T takes the form of 2-tuples;for the proposed ET-HLWA operator in this paper,the associated weight vector L =((l 1,g 1),(l 2,g 2),...,(l n ,g n ))T and the weight vector of 2-tuples C =((c 1,b 1),(c 2,b 2),...,(c n ,b n ))T take the form of 2-tuples.The THWA,T-HLWA and ET-HLWA operators aim at the arguments of 2-tuples while the LHAA operator aims at the arguments of virtual linguistic variables.
If 2-tuple linguistic variable is equivalent to the corresponding virtual linguistic term as stated by Dong et al.[24],then the THWA operator proposed in this paper is equivalent to the LHAA operator,but the LHAA operator can not deal with the weight vectors in the form of linguistic variables or 2-tuples while the T-HLWA and ET-HLWA operators can effectively solve this issue.From this point of view,the T-HLWA and ET-HLWA operators generalization the LHAA operator.
4.MAGDM method with 2-tuple linguistic assessments
4.1.MAGDM problem description using 2-tuple linguistic assessments This section describes the MAGDM problem with 2-tuple lin-guistic assessments.
Let A ={A 1,A 2,...,A m }be a discrete set of m possible alterna-tives and F ={a 1,a 2,...,a n }be a ?nite set of n attributes,where A i denotes the i th alternative and a j denotes the j th attribute.Let D ={D 1,D 2,...,D t }be a ?nite set of t experts,where D k denotes the k th expert.
The expert D k provides his/her assessment information of an
alternative A i on an attribute a j as a 2-tuple r k ij ?s k ij ;a k
ij according
to a prede?ned linguistic term set S ,where s k ij 2S ;a k
ij 2?à0:5;0:5T
ei ?1;2;...;m ;j ?1;2;...;n T.Thus,the experts’assessment infor-mation can be represented by the 2-tuple linguistic decision matri-ces R k
?r k ij
m ?n
ek ?1;2;...;t T.Suppose that both attribute weights and expert weights also can be represented by the 2-tuple linguistic information.Let W =((w 1,h 1),-...,(w n ,h n ))T be the 2-tuple linguistic weight vector of the attributes a j (j =1,2,...,n )and C =((c 1,b 1),(c 2,b 2),...,(c t ,b t ))T be the 2-tuple linguistic weight vector of the experts D k (k =1,2,...,t ),where w j 2S ,c k 2S ,h j 2[à0.5,0.5)and b k 2[à0.5,0.5).
The problem concerned in this paper is how to rank alternatives or select the most desirable alternative(s)among the ?nite set A on the basis of the 2-tuple linguistic decision matrices and the 2-tuple linguistic weight information of attributes and experts.4.2.The decision method with 2-tuple linguistic assessments
In this section,we propose a new method to solve the MAGDM problems with 2-tuple linguistic assessments.An algorithm and process of the MAGDM problems with 2-tuple linguistic assess-ments may be given as follows.
Step 1.Utilized the decision matrix R k and the ET àWA operator,
the individual overall preference value z k i ?s k i ;a k i àá
of the alternative A i is derived as follows:
z k i ?s k i ;a k i àá?ET àWA W s k i 1;a k i 1
àá;...;s k in ;a k in àáàá?D X n j ?1D à1ew j ;h j TP j ?1D ew j ;h j TD à1s k ij ;a k ij !;s k i 2S ;a k
i
2?à0:5;0:5T;
e12T
where W =((w 1,h 1),...,(w n ,h n ))T be the 2-tuple linguistic weight
vector of the attributes a j (j =1,2,...,n ).
Step https://www.doczj.com/doc/d7760242.html,ed the ET àHLWA operator to integrate all the individ-ual overall preference value z k i ?s k i ;a k i àá
ek ?1;2;...;t Tof alternative A i ,the collective overall preference value z i =(s i ,a i )of alternative A i is obtained as follows:
z i ?es i ;a i T?ET àHLWA L ;C s 1i ;a 1
i àá;...;s t i ;a t i
àáàá?D X t j ?1D à1el j ;g j T
P j ?1D el j ;j T
D à1s 0r ej Ti ;a 0r ej Ti !;e13T
where L =((l 1,g 1),(l 2,g 2),...,(l t ,g t ))T is the 2-tuple linguistic
weighted vector correlating with ET àHLWG ;es 0r ej Ti ;a 0r ej T
i Tis the j th largest 2-tuple of 2-tuples s 0k i ;a 0k i àáek ?1;2;...;t Twith s 0k i ;a 0k i àá?t ec k ;b k T s k i ;a k i àáàá
,and C =((c 1,b 1),(c 2,b 2),...,(c t ,b t ))T is the 2-tuple linguistic weight vector of experts D k (k =1,2,...,t ).Step 3.Rank all the alternatives A i (i =1,2,...,m )and select the
best one(s)in accordance with the 2-tuple z i =(s i ,a i )(i =1,2,...,m ).If any alternative has the highest z i value,then,it is the best alternative.
S.-P.Wan /Knowledge-Based Systems 45(2013)31–4035
Remark4.In Step2,we suppose that the weight vector of experts is in the form of2-tuples C=((c1,b1),(c2,b2),...,(c t,b t))T,so the ETàHLWA operator is used to integrate the individual overall pref-erence values of alternative into the collective one.Even if the weight vector of experts is in the form of linguistic labels,the ETàHLWA operator can also be used to obtain the collective one since the linguistic term can be readily transformed into2-tuple by Eq.(3).But if the weight vector of experts is in the form of real numbers,then the TàHLWA operator(i.e.,(10))can be used to obtain the collective one.
Remark 5.Obviously,real number form and2-tuple form are quite different for the weight vector of experts.In order to obtain the collective overall preference values of alternatives,only the ETàHLWA operator can be used for the latter while only the TàHLWA operator for the former.
Remark6.In this paper,we take the weight vector of experts as the2-tuple form.If the2-tuples for the weight vector of experts and for the assessment of attribute values come from different lin-guistic term sets,they should be?rstly converted into the fuzzy sets de?ned in the basic linguistic term set as stated in Remark 1.This solves the normalization problem of the2-tuple weight vec-tor of experts.
5.A real application to a personnel selection problem
In this section,a real personnel selection problem is used to illustrate the proposed method in this paper.
Ahead Software Company Limited was registered in Nanchang, Jiangxi of China.It is a key national project software enterprise and key national high-tech enterprise.Established in1994,it special-izes in research and develop of platform software and trade appli-cation software and selling.The company desires to hire a system analyst from national recruitment.The expert panel consists of two board members D1and D2,Company chairman D3and Company vice chairman D4.Since Company chairman D3has engaged in hu-man resource management for many years and accumulated rich experience,Company chairman D3is named as the group leader which is responsible for the whole recruitment work.
After preliminary screening,?ve candidates(i.e.,alternatives) A i(i=1,2,...,5)remain for further evaluation.Generally,many attributes should be used to evaluate these candidates.To effec-tively and rapidly make decision,three attributes are chosen by the four experts.These attributes are oral communication skills a1,emotional steadiness a2and past experience a3,respectively. Since these attributes are all qualitative attributes,it is reason-able for the experts to use linguistic variables or2-tuples to rep-resent the evaluation information of the candidates with respect to the attributes.Consequently,the?ve candidates are to be evaluated using the2-tuple linguistic information according to the linguistic term set:
S?f s0?extremely poor;s1?very poor;s2?poor;
s3?slightly poor;s4?fairemediumT;
s5?slightly goodeimportantT;s6?goodeimportantT;
s7?very goodeimportantT;s8?extremely goodeimportantTg
by the four experts under these three attributes.The2-tuple lin-guistic decision matrices provided by each expert are respectively as follows:
R1?
es0;0:4Tes3;0:2Tes8;0:1T
es4;0:3Tes1;0:4Tes7;à0:2T
es2;0:2Tes4;0:3Tes6;0:3T
es1;0:3Tes5;à0:4Tes7;0:2T
es7;à0:2Tes8;0:1Tes0;0:1T
B B
B B
B B
B B
@
1
C C
C C
C C
C C
A
;R2?
es2;0:1Tes4;0:2Tes6;0:1T
es5;à0:3Tes3;0:1Tes6;0:2T
es2;0:2Tes7;à0:3Tes6;0:3T
es2;0:3Tes1;0:4Tes7;0:2T
es6;0:2Tes7;à0:1Tes8;0:1T
B B
B B
B B
B B
@
1
C C
C C
C C
C C
A
;
R3?
es4;0:3Tes2;0:4Tes7;0:3T
es3;0:4Tes2;0:1Tes5;à0:2T
es1;0:3Tes4;0:3Tes6;0:3T
es5;0:1Tes8;à0:3Tes7;0:2T
es7;à0:2Tes7;0:4Tes2;0:4T
B B
B B
B B
@
1
C C
C C
C C
A
;R4?
es1;0:3Tes0;0:4Tes7;0:1T
es3;0:3Tes5;0:4Tes8;à0:2T
es1;0:2Tes6;0:2Tes8;0:3T
es1;0:4Tes5;0:3Tes8;à0:2T
es6;0:3Tes3;0:1Tes1;0:3T
B B
B B
B B
@
1
C C
C C
C C
A
:
With ever increasing complexity in real human resource manage-
ment,it is very dif?cult to give precisely the linguistic assessment
information on the expert weights and attribute weights according
to the given linguistic term set in advance.For example,the experts
think that the past experience a3is important and the weight may
be s6but less than s6,thus the weight of attribute a3can be repre-
sented using the linguistic2-tuple(w3,h3)=(s6,à0.2).After the
negotiation and investigation of the experts,they determine the
2-tuple linguistic weight vector W=((w1,h1),(w2,h2),(w3,h3))T of
the attributes,where(w1,h1)=(s8,à0.4),(w2,h2)=(s1,0.3)and
(w3,h3)=(s6,à0.2).
As the stated earlier,Company chairman D3,named as the
group leader,has rich experience,knowledge and speciality in hu-
man resource management.Obviously,his importance degree is
extremely high and may be s8but less than s8,therefore,the
weight of Company chairman D3can be represented using the lin-
guistic2-tuple(c3,b3)=(s8,à0.1).Analogously,the2-tuple linguis-
tic weight vector C=((c1,b1),(c2,b2),(c3,b3),(c4,b4))T of the experts
can be obtained,where(c1,b1)=(s5,0.1),(c2,b2)=(s1,0.2),
(c3,b3)=(s8,à0.1)and(c4,b4)=(s3,0.4).
Next,we adopt the proposed method to solve the above person-
nel selection example.
https://www.doczj.com/doc/d7760242.html,bine the decision matrix R1and the2-tuple linguistic
weight vector of attributes W=((w1,h1),(w2,h2),(w3,h3))T
with the ETàWA operator,the individual overall prefer-
ence value z1
1
?s1
1
;a11
àá
of candidate A1is derived as
follows:
z1
1
?s1
1
;a11
àá
?ETàWA W s1
11
;a111
àá
;s1
12
;a112
àá
;s1
13
;a113
àá
àá
?D
X3
j?1
Dà1ew j;h jT
P3
j?1
Dà1ew j;h jT
Dà1s1
1j
;a11j
!
?es4;à0:3143T:
Likewise,we have
z1
2
?s1
2
;a12
àá
?es5;0:0299T;z1
3
?s1
3
;a13
àá
?es4;0:0034T;z1
4
?s1
4
;a14
àá
?es4;à0:0803T;z1
5
?s1
5
;a15
àá
?es4;0:2714T:
z2
1
?s2
1
;a21
àá
?es4;à0:1361T;z2
2
?s2
2
;a22
àá
?es5;0:1503T;z2
3
?s2
3
;a23
àá
?es4;0:2156T;z2
4
?s2
4
;a24
àá
?es4;0:1537T;z2
5
?s2
5
;a25
àá
?es7;0:0116T:
z3
1
?s3
1
;a31
àá
?es5;0:3156T;z3
2
?s3
2
;a32
àá
?es4;à0:1626T;
z3
3
?s3
3
;a33
àá
?es4;à0:4619T;z3
4
?s3
4
;a34
àá
?es6;0:1585T;
z3
5
?s3
5
;a35
àá
?es5;0:1170T:
z4
1
?s4
1
;a41
àá
?es4;à0:4912T;z4
2
?s4
2
;a42
àá
?es5;0:2612T;
z4
3
?s4
3
;a43
àá
?es4;0:4435T;z4
4
?s4
4
;a44
àá
?es4;0:2701T;
z4
5
?s4
5
;a45
àá
?es4;0:0442T:
Step2.Assume that the correlated2-tuple weighted vector with
ETàHLWA operator is L=((l1,g1),(l2,g2),(l3,g3),(l4,g4))T,
where(l1,g1)=(s2,0.2),(l2,g2)=(s5,0.1),(l3,g3)=(s7,à0.2),
and(l4,g4)=(s6,0.3).Used the2-tuple linguistic weight
vector of experts C=((c1,b1),(c2,b2),(c3,b3),(c4,b4))T and
36S.-P.Wan/Knowledge-Based Systems45(2013)31–40
the ETàHLWA operator to integrate all the individual
overall preference values z k
1?s k
1
;a k1
àá
(k=1,2,3,4)of can-
didate A1,the collective overall preference value of candi-date A1is thus calculated as follows:
z1?ETàHLWA L;C s1
1;a11
àá
;...;s4
1;a41
àáàá
?D
X4
j?1
Dà1el j;g jT
P4
j?1
Dà1el j;g jT
Dà1s0rejT
1
;a0rejT
1
!
?es4;0:2189T:
Similarly,we have
z2?ETàHLWA L;C s1
2;a12
àá
;...;s4
2;a42
àáàá
?D
X4
j?1
Dà1el j;g jT
P
j?1
Del j;
j
T
Dà1es0rejT
2
;a0rejT
2
T
!
?es5;à0:2661T;
z3?ETàHLWA L;C s1
3;a13
àá
;...;s4
3;a43
àáàá
?D
X4
j?1
Dà1el j;g jT
P4
j?1
Dà1el j;g jT
Dà1s0rejT
3
;a0rejT
3
!
?es4;0:0373T;
z4?ETàHLWA L;C s1
4;a14
àá
;...;s4
4;a44
àáàá
?D
X4
j?1
Dà1el j;g jT
P4
j?1
Dà1el j;g jT
Dà1es0rejT
4
;a0rejT
4
T
!
?es5;à0:1673T;
and
z5?ETàHLWA L;C s1
5;a15
àá
;...;s4
5;a45
àáàá
?D
X4
j?1
Dà1el j;g jT
P
j?1
Del j;
j
T
Dà1s0rejT
5
;a0rejT
5
!
?es5;0:1682T:
Step3.Since z5>z4>z3>z1>z2,the ranking result of the candi-dates is A51A41A31A11A2and therefore the best can-
didate is A5,which will be recommended to Ahead
Software Company Limited.
https://www.doczj.com/doc/d7760242.html,parison analyses of the results obtained
https://www.doczj.com/doc/d7760242.html,parison with the approach to MAGDM with linguistic power average operators
Xu et al.[25]proposed four approaches to MAGDM with lin-guistic power average operators.To illustrate the superiorities of the proposed method,we use Approach I of[25]to solve the above personnel selection problem,and then conduct a comparison anal-ysis.The following symbols Sup kh,T k and V k see[25]in detail. Step1:Calculate the matrices Sup kh(k,h=1,2,3,4,k–h)as follows:
Sup12?Sup21?
0:78750:87500:7500 0:95000:78750:9250 1:00000:70001:0000 0:87500:60001:0000 0:92500:85000
B B
B B
B B
B@
1
C C
C C
C C
C A
;
Sup13?Sup31?
0:51250:90000:9000
0:88750:91250:7500
0:88751:00001:0000
0:52500:61251:0000
1:00000:91250:7125
B B
B B
B B
B@
1
C C
C C
C C
C A
;
Sup14?Sup41?
0:88750:65000:8750
0:87500:50000:8750
0:87500:76250:7500
0:98750:91250:9250
0:93750:37500:8500
B B
B B
B@
1
C C
C C
C A;
Sup23?Sup32?
0:72500:77500:8500
0:83750:87500:8250
0:88750:70001:0000
0:65000:21251:0000
0:92500:93750:2875
B B
B B
B@
1
C C
C C
C A;
Sup24?Sup42?
0:90000:52500:8750
0:82500:71250:8000
0:87500:93750:7500
0:88750:51250:9250
0:98750:52500:1500
B B
B B
B B
@
1
C C
C C
C C
A
;
Sup34?Sup43?
0:62500:75000:9750
0:98750:58750:6250
0:98750:76250:7500
0:53750:70000:9250
0:93750:46250:8625
B B
B B
B B
@
1
C C
C C
C C
A
:
Step2:Calculate the matrices T k(k=1,2,3,4)as follows:
T1?
2:18752:42502:5250
2:71252:20002:5500
2:76252:46252:7500
2:38752:12502:9250
2:86252:13751:5625
B B
B B
B@
1
C C
C C
C A;
T2?
2:41252:17502:4750
2:61252:37502:5500
2:76252:33752:7500
2:41251:32502:9250
2:83752:31250:4375
B B
B B
B@
1
C C
C C
C A;
T3?
1:86252:42502:7250
2:71252:37502:2000
2:76252:46252:7500
1:71251:52502:9250
2:86252:31251:8625
B B
B B
B B
@
1
C C
C C
C C
A
;
T4?
2:41251:92502:7250
2:68751:80002:3000
2:73752:46252:2500
2:41252:12502:7750
2:86251:36251:8625
B B
B B
B B
@
1
C C
C C
C C
A
:
Suppose that the weight vector of experts is w=(0.2898,0.0682,
0.4489,0.1932)T.Utilize w to calculate the matrices V k(k=1,2,3,4)
as follows:
V1?
0:29790:29970:2798
0:29070:28860:3076
0:29010:29050:2974
0:31760:32330:2919
0:28990:29530:2772
B B
B B
B@
1
C C
C C
C A;
V2?
0:07010:07050:0659
0:06840:06790:0724
0:06830:06840:0700
0:07470:07610:0687
0:06820:06950:0652
B B
B B
B@
1
C C
C C
C A;
S.-P.Wan/Knowledge-Based Systems45(2013)31–4037
V3?
0:46150:46430:4335 0:45030:44700:4764 0:44940:45000:4607 0:49190:50080:4522 0:44910:45750:4294
B B
B B
B B
@
1
C C
C C
C C
A
;
V4?
0:19860:19980:1866 0:19380:19240:2050 0:19340:19370:1983 0:21170:21550:1946 0:19330:19690:1848
B B
B B
B B
@
1
C C
C C
C C
A
:
Step3:Utilize the LPWA operator to aggregate all the individual decision matrixes into the collective decision matrix as
follows:
R?
s2:5089s2:4495s7:1575
s3:7419s2:5921s6:4264
s1:6049s4:8426s6:8631
s3:3898s6:5916s7:3703
s6:6654s6:8677s1:8271
B B
B B
B B
@
1
C C
C C
C C
A
:
Step4:Suppose that the weight vector of attributes is x=(0.5170,0.0884,0.3946)T.Utilize x,R and the LWA
operator to derive the collective overall preference values
z i(i=1,2,3,4,5)of the alternatives as follows:
z1?s4:3380;z2?s4:6995;z3?s3:9660;z4?s5:2436;z5?s4:7741:
Thus,the ranking result obtained by[25]is A41A51A21A11A3, which is remarkably different from that obtained by this paper. The best alternative by the former is A4while that by the latter is A5.The worst alternative by the former is A3while that by the latter is A2.The main reasons and comparison analysis are made as follows:
(A)Xu et al.[25]proposed Approaches I and II to MAGDM based
on the LPWA and LPOWA operators,respectively.However, different experts assess the alternatives according to the same extended continuous linguistic term set in these approaches.In real-life decision problems,different experts may express their opinions from different granularity lin-guistic term sets.These approaches in[25]did not discuss this case while the proposed method in this paper can be used to solve multi-granularity linguistic MAGDM(as stated in Remarks1and6).
(B)The weighted vectors for LPWA and LPOWA operators are
only in the form of real numbers.That is to say,the four approaches to MAGDM with linguistic power average oper-ators proposed in[25]can only deal with the situation where the weight vectors of attributes and experts are all real numbers rather than linguistic variables or2-tuples.
Whereas,the method proposed in this paper can deal with three cases:linguistic variables,2-tuples and numerical val-ues for the weight information of attributes and experts(see Subsection6.2in detail),which are the notable difference between the method[25]and the method in this paper.
(1)If the weight information of experts is given by linguistic
variables,the linguistic variables can be easily trans-
formed into2-tuples by using Eq.(3),then the
ETàHLWA operator can still be used to integrate the
individual overall preference values of alternatives and
to derive the collective ones of alternatives(see the third
line of Table1in Subsection6.2).
(2)If the weight information of experts is given by2-tuples,
the ETàHLWA operator can be directly used to integrate
the individual overall preference values of alternatives
and to derive the collective ones of alternatives(see
the example of Section5).
(3)If the weight information of experts is given by the real
numbers,we can use the TàHLWA operator to replace
the ETàHLWA operator to derive the collective overall
preference values of alternatives.
For example,suppose that the weight vector of experts is w=(0.2898,0.0682,0.4489,0.1932)T,then used the ETàHLWA operator(i.e.,Eq.(10)),the collective overall preference values of alternatives are obtained as follows:
z1?es4;3953T;z2?es5;à0:4525T;z3?es4;à0:1060T;z4?es5;0:0082T;z5?es5;à0:2061T:
Hence,the ranking order of the alternatives is A41A51A21A1-1A3,which is accordance with that obtained by[25].
In sum,the above discussion demonstrates that the proposed method in this paper is of universality and?exibility.
https://www.doczj.com/doc/d7760242.html,parison with the best related2-tuple MAGDM method
Wei[22]proposed a MAGDM method based on the ET-WG and ET-OWG operators with2-tuple linguistic information.In the follow-ing,to further illustrate the superiorities of the proposed method,we use the method proposed in this paper to solve the investment selec-tion problem of[22],and then conduct a comparison analysis.
An investment company wants to invest a sum of money in the best option.There is a panel with?ve possible alternatives to in-vest the money:a car company A1,a food company A2,a computer company A3,an arms company A4and a TV company A5.The investment company must take a decision according to the four attributes:the risk analysis a1,the growth analysis a2,the social-political impact analysis a3and the environmental impact analysis a4.The?ve possible alternatives A i(i=1,2,3,4,5)are to be evalu-ated using the linguistic term set S={s1=extremely poor(EP);s2-=very poor(VP);s3=poor(P);s4=medium(M);s5=good(G); s6=very good(VG);s7=extremely good(EG)}by three experts D k(k=1,2,3)under the above four attributes.They respectively construct the decision matrices R k?~r k
ij
5?4
ek?1;2;3Tas follows: R1?
M G P P
P VP M P
G M G EP
VG P P G
EG EP VP M
B B
B B
B B
B B
@
1
C C
C C
C C
C C
A
;R2?
P M VP VP
VP EP G G
M G P EG
EG VP VP M
P VP M VP
B B
B B
B B
B B
@
1
C C
C C
C C
C C
A
;
R3?
G P VP VG
VP G P G
VG VP G P
G VG EG VP
M VP M G
B B
B B
B B
@
1
C C
C C
C C
A
:
In[22],the linguistic weight vector of the attributes is
H?es0
3
;s0
1
;s0
2
;s0
4
Tusing the linguistic term set S0?f s0
1
?
extremely important;s0
2
?very important;s0
3
?important;s0
4?medium;s0
5
?bad;s0
6
?very bad;s0
7
?extremely bad g.For the ETàOWG operator[22],the correlated linguistic weighted vector
is taken as V?s0
6
;s0
4
;s0
2
àáT
.
We suppose that the weight vector of experts is x?s04;s04;s05
àáT according to the linguistic term set S0.In addition,for the ETàHL-WA operator of this paper,we also take the correlated linguistic
weighted vector as V?s0
6
;s0
4
;s0
2
àáT
.
38S.-P.Wan/Knowledge-Based Systems45(2013)31–40
Applied the proposed method in this paper,the above linguistic decision matrices,the linguistic weight vectors of the attributes and experts,and the correlated linguistic weighted vector should be?rstly transformed into2-tuple linguistic forms by using Eq.
(3).Then,repeating the same steps as in Section5,the collective overall preference values of alternatives can be obtained.Table1 lists the collective overall preference values of alternatives ob-tained by the method[22]and method in this paper.
It is easily seen from Table1that the ranking results obtained by the method[22]and the method proposed in this paper are slightly different.The difference is the ranking order of A4and A3, i.e.,A41A3by the former while A31A4by the latter.The worst alternative is A2by both methods,but the best alternative by the former is A4,while the best alternative by the latter is https://www.doczj.com/doc/d7760242.html,-pared with the former,the main advantages of the latter mainly lie in the following:
(i)The latter suf?ciently considers the importance degrees of
different experts.Before utilizing the ETàHLWA operator, the individual overall preference values of alternatives should be?rst weighted by the expert weights and then the collective ones of alternatives can be obtained.However, the former is based on the ETàWG and ETàOWG operators, which does not consider the importance degrees of different experts at https://www.doczj.com/doc/d7760242.html,ly,the former supposed that the expert weights are absolutely unknown and used the ETàOWG operators to integrate the individual overall preference val-ues of alternatives into the collective ones.
In real-life decision problems,different experts usually act as different roles in the decision process(such as the expert D3in Section5).Some experts may assign unduly high or unduly low uncertain preference values to their preferred or repugnant objects.To relieve the in?uence of these unfair arguments on the decision results and re?ect the importance degrees of all the experts,the latter?rst weights each indi-vidual overall preference value by using the corresponding expert weight,and then utilizes the ETàHLWA operator to aggregate all the individual weighted overall preference val-ues of each alternative into the collective ones of alterna-tives.Therefore,the ETàHLWA or TàHLWA operator can make the decision results more reasonable through assign-ing low weights to those‘‘false’’or‘‘biased’’arguments.
These advantages can not be re?ected in the former.
(ii)The former is only suitable for the case where the weight information of attributes is the form of the linguistic vari-ables,whereas the latter can deal with the three cases:lin-guistic variables,2-tuples and numerical values for the weight information of attributes and experts(see Subsection
6.1in detail),which also shows that the latter is more uni-
versal and?exible than the former.
https://www.doczj.com/doc/d7760242.html,parison with other normal linguistic MAGDM methods
Ma et al.[31]developed a fuzzy multi-criteria group decision-making(MCGDM)support system,which is called a Decider.By means of existing works on linguistic methods,Ma et al.[31]can deal with subjective and objective information at the same time. The Comparison analyses between[31]and this paper are con-ducted from four aspects.
(a)The research focuses of both papers are quite different.The
former constructed a MCGDM model and developed a fuzzy MCGDM support system under multi-level criteria and multi-level evaluators,while the latter focuses on develop-ing some new2-tuple linguistic hybrid arithmetic aggrega-tion operators.
(b)Although the subjective information in the former may be in
the form of linguistic terms,the linguistic assessment infor-mation is all transformed into fuzzy numbers to deal with them.Any transformation process between linguistic terms and fuzzy numbers may easily result in information losses and distortions to some degree.However,the latter utilizes the proposed2-tuple linguistic hybrid arithmetic aggrega-tion operators to integrate the linguistic assessment infor-mation,which needs not such transformation between linguistic terms and fuzzy numbers.
(c)The former selected the existing aggregation operators
rather than developed news aggregation operators to inte-grate in the MCGDM model,whereas the latter developed some new2-tuple linguistic hybrid arithmetic aggregation operators and applied to the MAGDM.
(d)The former constructed the MCGDM model with multi-level
hierarchies of criteria and evaluators,while the MAGDM model in the latter has only one-level hierarchy of attributes.
Though the former can be applied to many decision problems, such as fabric material ranking,strategy evaluation and non-woven product assessment,then the latter also has some prominent advantages as stated in Subsection6.2in detail.
7.Conclusion
This paper de?ned the operation laws for2-tuples and studied the related properties of the operation laws.After reviewing the existing2-tuple linguistic arithmetic aggregation operators,some hybrid arithmetic aggregation operators with2-tuple linguistic information were developed including THWA,TàHLWA,and ETàHLWA operators.The THWA operator generalizes both the TWA and TOWA operators.The ETàOWA operator is a special case of the TàHLWA operator.
A new decision method was proposed to solve the MAGDM problem with2-tuple linguistic information.The method is based on ETàWA and ETàHLWA operators which can suf?ciently con-sider the importance degrees of different experts and thus relieve the in?uence of those unfair arguments on the decision results. The proposed hybrid arithmetic aggregation operators with2-tu-ple linguistic information enlarge the research content on2-tuple linguistic information and enrich the ideas for solving fuzzy MAG-DM problems with linguistic information.Although the developed method was illustrated using a personnel selection problem,it will be expected to be applicable to decision problems in many areas, especially in situations where multiple experts are involved and the weights of attributes and experts are represented by linguistic variables or2-tuples instead of real numbers,such as the enter-prise project selection and water environment assessment,partner choice in supply chain,and so on.Furthermore,the developed method can also deal with the expert weights in the form of real numbers only if we use the TàHLWA operator to replace the ETàHLWA operator,which indicates that the method proposed in this paper is of universality and?exibility.
In this paper we do not make any conclusion about the deter-mining method of the linguistic(or2-tuple linguistic)weighted vector correlating with2-tuple linguistic hybrid arithmetic aggre-gation operators and effectively determining the expert weights in the form of the linguistic or2-tuples,which will be investigated in the near future.In addition,2-tuple linguistic hybrid geometric aggregation operators are also worthy of consideration for future research.
Acknowledgments
This work was partially supported by the National Natural Sci-ence Foundation of China(Nos.71061006,61263018,71171055
S.-P.Wan/Knowledge-Based Systems45(2013)31–4039
and70871117),the Program for New Century Excellent Talents in University(the Ministry of Education of China,NCET-10-0020),the Specialized Research Fund for the Doctoral Program of Higher Edu-cation of China(No.20113514110009),the Humanities Social Sci-ence Programming Project of Ministry of Education of China(No. 09YGC630107),the Natural Science Foundation of Jiangxi Province of China(No.20114BAB201012)and the Science and Technology Project of Jiangxi province educational department of China(Nos. GJJ12265and GJJ12740)and the Excellent Young Academic Talent Support Program of Jiangxi University of Finance and Economics. References
[1]J.M.Tapia Garc?′a,M.J.del Moral,M.A.Mart?′nez, E.Herrera-Viedma,A
consensus model for group decision making problems with linguistic interval fuzzy preference relations,Expert Systems with Applications39 (2012)10022–10030.
[2]I.J.Pérez,F.J.Cabrerizo,E.Herrera-Viedma,Group decision making problems in
a linguistic and dynamic context,Expert Systems with Applications38(3)
(2011)1675–1688.
[3]J.Lu,G.Zhang, D.Ruan, F.Wu,Multi-Objective Group Decision Making:
Methods,Software and Applications with Fuzzy Set Techniques,Imperial College Press,London,2007.
[4]Z.S.Xu,A method based on linguistic aggregation operators for group decision
making with linguistic preference relations,Information Sciences166(2004) 19–30.
[5]Z.S.Xu,Deviation measures of linguistic preference relations in group decision
making,Omega33(3)(2005)249–254.
[6]Z.S.Xu,On method of multi-attribute group decision making under pure
linguistic information,Control and Decision19(7)(2004)778–781.
[7]L.G.Zhou,H.Y.Chen,A generalization of the power aggregation operators for
linguistic environment and its application in group decision making, Knowledge-Based Systems26(2012)216–224.
[8]Z.Zhang,C.H.Guo,A method for multi-granularity uncertain linguistic group
decision making with incomplete weight information,Knowledge-Based Systems26(2012)111–119.
[9]X.T.Wang,W.Xiong,An integrated linguistic-based group decision-making
approach for quality function deployment,Expert Systems with Applications 38(2011)14428–14438.
[10]Z.S.Xu,Uncertain linguistic aggregation operators based approach to multiple
attribute group decision making under uncertain linguistic environment, Information Sciences168(2004)171–184.
[11]F.J.Cabrerizo,I.J.Pérez,E.Herrera-Viedma,Managing the consensus in group
decision making in an unbalanced fuzzy linguistic context with incomplete information,Knowledge-Based Systems23(2)(2010)169–181.
[12]F.Herrera,E.Herrera-Viedma,L.Mart?′nez,A fuzzy linguistic methodology to
deal with unbalanced linguistic term sets,IEEE Transactions on Fuzzy Systems 16(2)(2008)354–370.
[13]F.J.Cabrerizo,I.J.Pérez,E.Herrera-Viedma,Managing the consensus in group
decision making in an unbalanced fuzzy linguistic context with incomplete information,Knowledge-Based Systems23(2010)169–181.
[14]X.H.Yu,Z.S.Xu,Q.Chen,A method based on preference degrees for handling
hybrid multiple attribute decision making problems,Expert Systems with Applications38(2011)3147–3154.[15]F.Herrera,L.Mart?′nez,P.J.Sánchez,Managing non-homogeneous information
in group decision-making,European Journal of Operational Research166(1) (2005)115–132.
[16]J.M.Merigo,M.Casanovs,L.Mart?′nez,Linguistic aggregation operator for
linguistic decision making based no the Dempster–Shafer theory of evidence, International Journal of Uncertainty,Fuzziness and Knowledge-Based Systems 18(3)(2010)287–304.
[17]F.Herrera,L.Mart?′nez,A2-tuple fuzzy linguistic representation model for
computing with words,IEEE Transactions on Fuzzy Systems8(2000)746–752.
[18]K.H.Chang,T.C.Wen,A novel ef?cient approach for DFMEA combining2-tuple
and the OWA operator,Expert Systems with Applications37(3)(2010)2362–2370.
[19]G.W.Wei,Some generalized aggregating operators with linguistic information
and their application to multiple attribute group decision making,Computers &Industrial Engineering61(1)(2011)32–38.
[20]Y.Zhang,Z.P.Fan,An approach to linguistic multiple attribute decision making
with linguistic information based on ELOWA operator,Systems Engineer24
(12)(2006)98–101.
[21]Z.Pei,D.Ruan,J.Liu,Y.Xu,A linguistic aggregation operator with three kinds
of weights for nuclear safeguards evaluation,Knowledge-Based Systems28 (2012)19–26.
[22]G.W.Wei,A method for multiple attribute group decision making based on the
ET-WG and ET-OWG operators with2-tuple linguistic information,Expert Systems with Application37(12)(2010)7895–7900.
[23]G.W.Wei,X.F.Zhao,Some dependent aggregation operators with2-tuple
linguistic information and their application to multiple attribute group decision making,Expert Systems with Applications39(2012)5881–5886. [24]Y.C.Dong,Y.F.Xu,H.Y.Li,B.Feng,The OWA-based consensus operator under
linguistic representation models using position indexes,European Journal of Operational Research203(2010)455–463.
[25]Y.J.Xu,Jose M.Merigo,H.M.Wang,Linguistic power aggregation operators and
their application to multiple attribute group decision making,Applied Mathematical Modelling36(11)(2012)5427–5444.
[26]F.Herrera,L.Mart?′nez,A model based on linguistic2-tuple for dealing with
multi-granular hierarchical linguistic contexts in multi-expert decisionmaking,IEEE Transactions on Systems,Man,and Cybernetics31 (2001)227–234.
[27]F.Herrera,E.Herrera-Viedma,L.Mart?′nez,A fusion approach for managing
multi-granularity linguistic term sets in decision-making,Fuzzy Sets and Systems114(2000)43–58.
[28]S.Gramajo,L.Mart?′nez,A linguistic decision support model for QoS priorities
in networking,Knowledge-Based Systems32(2012)65–75.
[29]G.W.Wei,R.Lin,X.F.Zhao,H.J.Wang,Models for multiple attribute group
decision making with2-tuple linguistic assessment information,International Journal of Computational Intelligence Systems3(3)(2010)315–324.
[30]G.W.Wei,Grey relational analysis method for2-tuple linguistic multiple
attribute group decision making with incomplete weight information,Expert Systems with Application38(5)(2011)7895–7900.
[31]J.Ma,J.Lu,G.Q.Zhang,Decider:a fuzzy multi-criteria group decision support
system,Knowledge-Based Systems23(2010)23–31.
[32]Z.S.Xu,A method for multiple attribute decision making with incomplete
weight information in linguistic setting,Knowledge-Based Systems20(2007) 719–725.
[33]Z.S.Xu,A note on linguistic hybrid arithmetic averaging operator in group
decision making with linguistic information,Group Decision and Negotiation 15(2006)593–604.
40S.-P.Wan/Knowledge-Based Systems45(2013)31–40
