Journal of Intelligent&Fuzzy Systems26(2014)1601–1617
DOI:10.3233/IFS-130841
IOS Press
1601
Some new hybrid weighted aggregation operators under hesitant fuzzy multi-criteria decision making environment
Huchang Liao a and Zeshui Xu a,b,∗
a Antai College of Economics and Management,Shanghai Jiao Tong University,Shanghai,China
b College of Sciences,PLA University of Science and Technology,Nanjing,China
Abstract.Hesitant fuzzy t,as a new generalized type of fuzzy t,is an efficient and powerful structure in expressing uncertainty and vagueness and has attracted more and more scholars’attention.The aim of this paper is to develop some new aggregation operators to fu hesitant fuzzy information.The hesit
ant fuzzy hybrid arithmetical averaging(HFHAA)operator,the hesitant fuzzy hybrid arithmetical geometric(HFHAG)operator,the quasi HFHAA operator and the quasi HFHAG operator are propod and their properties are investigated.On the basis of the propod operators,some algorithms are introduced to aid multi-criteria single person decision making and multi-criteria group decision making respectively.Some examples are provided to illustrate the practicality and validity of our propod procedures.
Keywords:Group decision making,hesitant fuzzy t,hybrid weighted aggregation operator,multi-criteria decision making
1.Introduction
Since it was originally introduced by Zadeh[1], the fuzzy t has turned out to be one of the most efficient decision aid techniques providing the ability to deal with uncertainty and vagueness.In realistic decision making,imprecision may ari due to the unquantifiable information,incomplete information, unobtainable information,partial ignorance,and so forth[2].To cope with imperfect and impreci infor-mation whereby two or more sources of vagueness appear simultaneously,Zadeh’s traditional fuzzy t shows some limitations[3].The traditional fuzzy t us a crisp number in unit in
terval[0,1]as a member-ship degree of an element to a t;however,very often, such a crisp number is difficult to be determined for the ∗Corresponding author.Zeshui Xu,Tel./Fax:+862584483382; E-mails:xuzeshui@263;(H.Liao).decision maker.On the other hand,if a group of decision makers are asked to evaluate the candidate alternatives, they oftenfind some disagreements among themlves. Since the decision makers may have different opinions over the alternatives and they can’t persuade each other easily,a connsus result is hard to be obtained but a t of possible values.In such ca,the traditional fuzzy t also can not be ud to depict the group’s opinions. Hence,the classical fuzzy t has been extended into veral different forms,such as the intuitionistic fuzzy t[4],the interval-valued intuitionistic fuzzy t[5], the type2fuzzy t[6],the type n fuzzy t[7],the fuzzy multits(also named the fuzzy bags)[8],and so on[9].All the extensions are bad on the same rationale that it is not clear to assign the membership degree of an element to afixed t[10].Recently,on the basis of the above extensional forms of the fuzzy t, Torra and Narukawa[10,11]propod a new general-ized type of fuzzy t called hesitant fuzzy t(HFS),
1064-1246/14/$27.50©2014–IOS Press and the authors.All rights rerved
1602H.Liao and Z.Xu/Some new hybrid weighted aggregation operators
which opens new perspectives for further rearch on decision making under hesitant environments.
Up to now,the HFS has attracted more and more scholars’attention[3,10–18],which in turn has shown the definite advantages of the HFS over all the other extended forms of fuzzy t.Torra[11]firstly gave the concept of HFS and defined some of its basic opera-tions.After that,Torra and Narukawa[10]prented an extension principle permitting to generalize the exist-ing operations on fuzzy ts to HFSs,and described the application of this new type of ts in the framework of group decision making.Xia et al.[12,13]defined some operators and gave an intensive study on hesitant fuzzy information aggregation techniques and their applica-tion in decision making.Zhu and Xu[14,15]propod the hesitant fuzzy Bonferroni means and hesitant fuzzy geometric Bonferroni means for multi-criteria decision making with hesitant fuzzy information.Subquently, Xu and Xia[16,17]investigated the distance,sim-ilarity,and correlation measures for HFSs.Wei and Zhao[18]investigated multiple attribute decision mak-ing with hesitant interval-valued fuzzy information and propod some new Einstein aggregation operators with hesitant interval-valued fuzzy information.Combining intuitionistic fuzzy t and HFS,Zhu et al.[19]pro-pod dual hesitant fuzzy t(DHFS)as an extension of HFS,which consists of two ,the mem-bership hesitancy function and the non-membership hesitancy function.Wei et al.[20]investigated the multiple attribute
decision making(MADM)problems in which attribute values take the form of hesitant triangular fuzzy information,and developed some hes-itant triangular fuzzy aggregation operators.In order to apply HFS to multi-criteria decision making,Liao et al.[3]investigated the VIKOR method within the context of hesitant fuzzy circumstances and they[21] also studied the multiplicative consistency of a hesi-tant fuzzy preference relation and the group connsus among different decision makers.Later on,to maker a more reasonable decision,Liao and Xu[22]propod a satisfaction degree bad interactive decision mak-ing method to derive the weights of the hesitant fuzzy preference relation.Zhang and Wei[23]introduced a Shapley valued-bad VIKOR method for multi-criteria decision making with hesitant fuzzy decision making. Wei et al.[24]investigated the hesitant fuzzy multi-ple attribute decision making with incomplete weight information and developed some optimization models to derive the weights.In order to handle qualitative ttings occurred in decision making,Rodriguez et al.
[25]introduced the concept of hesitant fuzzy linguis-tic term t(HFLTS)who envelope is an uncertain linguistic variable[26]and prented a multi-criteria linguistic decision making model in which the decision makers provide their asssments by eliciting linguistic expressions.Zhu and Xu[27]propod some consis-tency measures for hesitant fuzzy linguistic preference relations.
The aim of this paper is to investigate the aggrega-tion operators for HFSs.In[12],Xia and Xu developed a family of operators to fu hesitant fuzzy informa-tion,such as the hesitant fuzzy weighted averaging (HFW A)operator,the hesitant fuzzy weighted geo-metric(HFWG)operator,the hesitant fuzzy ordered weighted averaging(HFOW A)operator,and the hes-itant fuzzy ordered weighted geometric(HFOWG) operator.The HFW A and HFWG operators can be ud to weight the hesitant fuzzy arguments,but ignore the importance of the ordered position of the argu-ments,while the HFOW A and HFOWG operators only weight the ordered position of each given argument, but ignore the importance of the arguments.To solve this drawback,the hesitant fuzzy hybrid averaging (HFHA)operator and the hesitant fuzzy hybrid geo-metric(HFHG)operator were propod to aggregate hesitant fuzzy arguments,which weight all the given arguments and their ordered positions simultaneously. Hence,the two operators have many advantages than the above mentioned operators in aggregating hesitant fuzzy information.However,the two operators do not satisfy the basic property named idempotency,which is desirable for aggregating afinite collection of HFSs. Therefore,in this paper,we intend to develop some new hesitant fuzzy hybrid weighted aggregation oper-ators.In addition,inspired by the quasi hesitant fuzzy ordered weighted averaging(QHFOWA)operator pro-pod in[13],we extend our propod operator to more general forms.Consider the powerfulness of HFS in multi-criteria decision making,we also give some pro-ced
ures with our propod operators for multi-criteria single person decision making and multi-criteria group decision making.
To do so,the remainder of this paper is t out as follows.Section2gives some basic knowledge of HFS and the aggregation operators.Section3develops two new hesitant fuzzy hybrid weighted aggregation opera-tors named hesitant fuzzy hybrid arithmetical averaging (HFHAA)operator and hesitant fuzzy hybrid arithmeti-cal geometric(HFHAG)operator.Section4extends the HFHAA and HFHAG operators to the quasi HFHAA and the quasi HFHAG operators,respectively.In Sec-tion5,we apply our propod operators to single
H.Liao and Z.Xu/Some new hybrid weighted aggregation operators1603 person decision making and multi-criteria group deci-
sion making under hesitant fuzzy environments.Some
practical examples are provided to illustrate the u of
our propod procedures.The paper ends with some
conclusions in Section6.
2.Some basic concepts and hesitant fuzzy
aggregation operators
Hesitant fuzzy t[10,11],as a generalization of
fuzzy t,permits the membership degree of an element
to a t prented as veral possible values between0
and1,which can better describe the situations where
people have hesitancy in providing their preferences
over objects in the process of decision making.To
facilitate our prentation,in what follows,let’s review
some basic concepts and hesitant fuzzy aggregation
operators.
Definition1.[10,11]Let X be afixed t,a hesitant
fuzzy t(HFS)on X is in terms of a function that
when applied to X returns a subt of[0,1],which can
be reprented as the following mathematical symbol:
E={<x,h E(x)>|x∈X}(1)
where h E(x)is a t of some values in[0,1],denoting
the possible membership degrees of the element x∈X
to the t E.For convenience,Xia and Xu[12]called
h E(x)a hesitant fuzzy element(HFE)and H the t of
all the HFEs.
Torra and Narukawa[10]defined the complement,
union and interction about HFEs.Bad on that,Xia
and Xu[12]gave some operational laws on the HFEs
h,h1and h2:
Definition2.[12]Let h,h1and h2be three HFEs,and
λbe a positive real number,then
(1)hλ=∪
γ∈h
γλ
;
(2)λh=∪
γ∈h
1−(1−γ)λ
;
(3)h1⊕h2=∪
γ1∈h1,γ2∈h2{γ1+γ2−γ1γ2};
(4)h1⊗h2=∪
γ1∈h1,γ2∈h2
{γ1γ2};
(5)
n
⊕
j=1
h j=∪
γj∈h j
1−
n
j=1
(1−γj)
;
bole(6)
n
⊗
j=1
h j=∪
γj∈h j
n
j=1
γj
.
Note that the number of values in different HFEs
may be different.Let l h
E(x)
be the number of values in
h E(x).For two HFEs h1and h2,let l=max{l h
1
,l h
2
}.
the mass什么意思To operate correctly,Xu and Xia[12]gave the follow-geography是什么意思
ing regulation,which is bad on the assumption that
all the decision makers are pessimistic:If l h
1
<l h
2
,
then h1should be extended by adding the minimum
value in it until it has the same length with h2;if
l h
1
>l h
2
,then h2should be extended by adding the
minimum value in it until it has the same length
with h1.In this paper,we shall extend the shorter
one by adding the value of0.5in it,that is to say,
we assume that all the decision makers are compromi.
amphitheatre
Definition3.[12]For a HFE h,s(h)=1l
h
γ∈h
γis
called the score function of h,where l h is the number of
values in h.For two HFEs h1and h2,if s(h1)>s(h2),
then h1>h2;if s(h1)=s(h2),then h1=h2.
Definition 4.[21]For a HFE h,v(h)=
1
l h
γi,γj∈h(γi−γj)2is called the variance function
of h,where l h is the number of values in h,and v(h)is
called the variance degree of h.For two HFEs h1and
h2,if v(h1)>v(h2),then h1<h2;if v(h1)=v(h2),
then h1=h2.
Bad on the score function s(h)and the variance
function v(h),the comparison scheme can be developed
to rank any HFEs:
If s(h1)<s(h2),then h1<h2;
If s(h1)=s(h2),then
1)If v(h1)<v(h2),then h1>h2;
2)If v(h1)=v(h2),then h1=h2.
In order to export the operations on fuzzy ts
to HFSs,Torra and Narukawa[10]propod an
aggregation principle for HFEs:
Definition5.[10]Let E={h1,h2,...,h n}be a t of
n HFEs, be a function on E, :[0,1]n→[0,1],then
E=∪γ∈{h
1×h2×···×h n}
{ (γ)}(2)
Bad on the above extension principle,Xia and
Xu[12]developed a ries of specific aggregation
operators for HFEs:
Definition 6.[12]Let h j(j=1,2,...,n)be a
collection of HFEs,ω=(ω1,ω2,...,ωn)T be the
aggregation-associated vector such thatωj∈[0,1]and
n
j=1
ωj=1,then
1604H.Liao and Z.Xu /Some new hybrid weighted aggregation operators
1)A hesitant fuzzy weighted averaging (HFW A)operator is a mapping HFWA :H n →H ,such that
HFW A (h 1,h 2,...,h n )=n ⊕j =1
w j h j
=∪γ1∈h 1,γ2∈h 2,...,γn ∈h n 1−
n
blood is thicker than water
j =1
(1−γj )
w j
(3)
2)A hesitant fuzzy weighted geometric (HFWG)operator is a mapping HFWG :H n →H ,where
HFWG (h 1,h 2,...,h n )=n
⊗j =1h w
screencapturej j =∪γ1∈h 1,γ2∈h 2,...,γn ∈h n
n
j =1γw j j
(4)
In the ca where w =(1/n,1/n,...,1/n )T ,the
HFWA operator reduces to the hesitant fuzzy averaging (HFA)operator and the HFWG operator reduces to the hesitant fuzzy geometric (HFG)operator.
Bad on the idea of the ordered weighted averaging (OW A)operator [28],the following operators can be defined:
Definition 7.[12]Let h j (j =1,2,...,n )be a col-lection of HFEs,h σ(j )be the j th largest of them,ω=(ω1,ω2,...,ωn )T be the aggregation-associated vector such that ωj ∈[0,1]and n j =1ωj =1,then 1)A hesitant fuzzy ordered weighted averaging (HFOW A)operator is a mapping HFOW A:H n →H ,where
HFOW A (h 1,h 2,...,h n )=n ⊕j =1
ωj h σ(j )
=∪γσ(1)∈h σ(1),γσ(2)∈h σ(2),...,γσ(n )∈h σ(n )
× 1− n
j =1
(1−γσ(j ))
ωj
(5)世界末日 英语
2)A hesitant fuzzy ordered weighted geometric
(HFOWG)operator is a mapping HFOWG:H n →H ,where
HFOWG (h 1,h 2,...,h n )=n
⊗j =1
h ω
j
σ(j )
=∪γσ(1)∈h σ(1),γσ(2)∈h σ(2),...,γσ(n )∈h σ(n )
n
j =1
γωj σ(j )
(6)
In the ca where ω=(1/n,1/n,...,1/n )T
,the HFOW A operator reduces to the HFA operator,and the HFOWG operator becomes the HFG operator.
It is noted that the HFW A and HFWG operators only weight the hesitant fuzzy arguments themlves,but ignore the importance of the ordered position
of the arguments,while the HFOW A and HFOWG operators only weight the ordered position of each
given arguments,but ignore the importance of the arguments.To solve this drawback,Xia and Xu [12]then introduced some hybrid aggregation operators for hesitant fuzzy arguments,which weight all the given arguments and their ordered positions.
Definition 8.[12]For a collection of HFEs h j (j =
1,2,...,n ),λ=(λ1,λ2,...,λn
)T is the weight vec-tor of them with λj ∈[0,1]and
n j =1λj =1,n is the balancing coefficient which plays a role of balance,then we define the following aggregation operators,which are all bad on the mapping H n →H with an
aggregation-associated vector ω=(ω1,ω2,...,ωn )T
such that ωj ∈[0,1]and n j =1ωj =1:
1)The hesitant fuzzy hybrid averaging (HFHA)operator:
HFHA (h 1,h 2,...,h n )=n ⊕j =1
ωj ˙h
σ(j ) =∪˙γσ(1)∈˙h σ(1),˙γσ(2)∈˙h σ(2),...,˙γσ(n )∈˙h
σ(n )× 1− n
j =1
(1−˙γ
σ(j ))ωj
(7)
where ˙h
σ(j )is the j th largest of ˙h =nλk h k ,(k =1,2,...,n ).
2)The hesitant fuzzy hybrid geometric (HFHG)operator:
money是什么HFHG (h 1,h 2,...,h n )=n
⊗j =1¨h ωj σ(j )
=∪¨γσ(1)∈¨h σ(1),¨γσ(2)∈¨h σ(2),...,¨γσ(n )∈¨h
σ(n ) n
j =1
¨γωj σ(j )
(8)
where ¨h σ(j )is the j th largest of ¨h k =h nλk k ,(k =
1,2,...,n ).
Especially,if w =(1/n,1/n,...,1/n )T ,then the HFHA operator reduces to the HFOW A operator,the HFHG operator reduces to the HFOWG operator.
3.Some new hesitant fuzzy hybrid weighted aggregation operators Although the HFHA (HFHG)operator generalizes both the HFW A (HFWG)and HFOWA (HFOWG)operators by weighting
the given importance and the ordered position of the arguments,there is a flaw that the operator does not satisfy the desirable ,idempotency.An example can be ud to illustrate this drawback.
H.Liao and Z.Xu /Some new hybrid weighted aggregation operators 1605
Example 1.Assume h 1={0.3,0.3,0.3},h 2={0.3,0.3,0.3}and h 3={0.3,0.3,0.3}are three HFEs,who weight vector is λ=(1,0,0)T ,and the aggregation-associated vector is also ω=(1,0,0)T .Then,
˙h
1=3×1⊗h 1=3h 1=
1−(1−0.3)3,1−(1−0.3)3,1−(1−0.3)3
=(0.657,0.657,0.657);
˙h
2=3×0⊗h 2=0⊗h 2=
1−(1−0.3)0,1−(1−0.3)0,1−(1−0.3)0
=(0,0,0);
˙h
3=3×0⊗h 3=0⊗h 3=
1−(1−0.3)0,1−(1−0.3)0,1−(1−0.3)0
=(0,0,0).
Obviously,s ˙h
1 >s ˙h
2 =s ˙h
3 .By using Equation (7),we have HFHA (h 1,h 2,h 3)=3⊕j =1
ωj ˙h
σ(j ) =∪˙γσ(1)∈˙h σ(1),˙γσ(2)∈˙h σ(2),˙γσ(3)∈˙h
σ(3)×
1−(1−˙γ
σ(1))1(1−˙γσ(2))0(1−˙γσ(3))0 =(0.657,0.657,0.657)/={0.3,0.3,0.3}Analogously,¨h
1=h 3×1
1
=
h 31
=
0.33,0.33,0.3
3
=(0.027,0.027,0.027);
¨h 2=h 3×02=h 02=
0.30,0.30,0.30 =(0,0,0);
¨h 3=h 3×03=h 03=
0.30,0.30,0.30 =(0,0,0).HFHG (h 1,h 2,h 3)=3
⊗j =1¨h ωj σ(j )
=∪¨γσ(1)∈¨h σ(1),¨γσ(2)∈¨h σ(2),¨γσ(3)∈¨h σ(3) ¨γ1σ(1)¨γ0σ(2)¨γ0σ(3)
=(0,0,0)/={0.3,0.3,0.3}
Since idempotency is the most important property
for every aggregation operators [29]but the HFHA and HFWG operators don’t meet this basic property,we need to develop some new hybrid aggregation opera-tors which also weight the importance of each argument and its ordered position simultaneously.In this c-tion below,we focus
on solving this problem and try to develop some new hybrid operators for HFSs.
Consider the HFOW A operator given as Equation (5),it is equivalent to the following form:
HFOW A (h 1,h 2,...,h n )=n ⊕j =1
ωε(j )h j
(9)
where h j be the ε(j )th largest element of h j (j =1,2,...,n ).Inspired by this,supposing the weighting vector of the elements is λ=(λ1,λ2,...,λn )T ,in order to weight the position and the element simul-taneously,we can u such a form as n
⊕j =1λj ωε(j )h j ,
which weights both the position and the element.After normalization,a new hesitant fuzzy hybrid arithmetical averaging operator can be generated.
Definition 9.For a collection of HFEs h j (j =1,2,...,n ),a hesitant fuzzy hybrid arithmetical aver-aging (
HFHAA)operator is a mapping HFHAA:H n →H ,defined by an associated weighting vector ω=(ω1,ω2,...,ωn )T
with ωj ∈[0,1]and n j =1ωj =1,such that
HFHAA (h 1,h 2,...,h n )=n
⊕j =1
λj ωε(j )h j n
j =1λj ωε(j )
,(10)
where ε:{1,2,...,n }→{1,2,...,n }is the per-mutation such that h j is the ε(j )th largest element
墨西哥婚礼of the collection of HFEs h j (j =1,2,...,n ),and λ=(λ1,λ2,...,λn )T is the weighting vector of the HFEs h j (j =1,2,...,n ),with λj ∈[0,1]and n j =1λj =1.Theorem 1.For a collection of HFEs h j (j =1,2,...,n ),the aggregated value by using the HFHAA operator is also a HFE,and
HFHAA (h 1,h 2,...,h n )
=
γ1∈h 1,γ2∈h 2,...,γn ∈h n
×⎧⎨⎩
1−n j =1
(1−γj )λj ωε(j )
lindsayn j =1
λj ωε(j )
⎫⎬⎭,(11)
