International Journal of Fuzzy Systems, Vol. 16, No. 2, June 2014
184
A Generalized Multiple Attributes Group Decision Making Approach
Based on Intuitionistic Fuzzy Sets
Zhifu Tao, Huayou Chen, Ligang Zhou, and Jinpei Liu
Abstract1
The aim of this paper is to investigate an intuitionistic fuzzy sets based generalized multiple attributes
group decision making (GMAGDM) adapting to the
situation that the attribute sets considered by a group
of experts are not the same and the decision information are provided with intuitionistic fuzzy numbers
(IFNs). Firstly, we develop three general procedures
to handle different intuitionistic fuzzy sets based
GMAGDM issues with diverse weight information:
completely known, partly known and completely unknown. Then, a novel procedure on the basis of information collection and transformation is put forward. Therein the transformation relation between
the IFN and the interval-valued hesitant fuzzy element (IVHFE) is utilized. Finally, an investment selection problem is illustrated to show the reasonability and efficiency of the proposed algorithms.
Keywords: Multi-attributes group decision making,
generalized multi-attributes group decision making,
intuitionistic fuzzy sets, interval-valued hesitant fuzzy
element, information transformation.
1. Introduction
Decision making problem is widespread in real life,
and the most discussed decision situation is the
multi-attributes group decision making (MAGDM) issues. A MAGDM problem is to find the best from possible alternative sets X  {x1, x2 , , xm} via the decision information about attributes A={a1, a2, , an} and evaluation values given by a group of experts E={e1, e2, , el}.
Especially, a MAGDM problem will degenerated to be a
MADM problem while the number of experts l=1.
Corresponding Author: H.Y. Chen is with the School of Mathematical
Science, Anhui University, 230601, China.
E-mail: [email protected]
Z. F. Tao is with the School of Mathematical Science, Anhui University, Hefei Anhui, E-mail: [email protected].
L.G. Zhou is with the School of Mathematical Science, Anhui University.
J. P. Liu is with the School of Business, Anhui University.
Manuscript received 26 Nov. 2013; revised 7 April 2014; accepted 27
April 2014.
When dealing with group decision making, it’s necessary to consider the diverse types of uncertainty. Fuzzy
set theory and its natural ability to deal with uncertainty
could provide the needed flexibility to handle the uncertainty factors in decision making. Thus, it’s of practical
meaning to study such a kind of MAGDM problem under fuzzy environment. Intuitionistic fuzzy set (IFS) was
presented by Atanassov [1] based on Zadeh’s fuzzy set
[2], which has been proofed to be a very useful tool. The
interval-valued fuzzy set (IVFS) [3] and the hesitant
fuzzy set (HFS) [4-5] are also two generalizations of the
fuzzy set. But in nature, IFS and IVFS, IVFS and HFS
are equivalent, respectively [4-6].
The application of IFS in MAGDM problem is a hot
topic in recent years [4, 7-13], among which the decision
information provided with IFSs and intuitionistic fuzzy
preference relations are two main fields. A fuzzy consensus discussion on the basis of the distance was given
by Szmidt and Kacprzyk [7]. Xu [8] discussed some
types of intuitionistic preference relations and their
properties, then provided an application of such information in the MAGDM issue. Wang [9] derived the intuitionistic fuzzy weights from intuitionistic fuzzy preference relations by building some linear goal programming models. Atanassov, Pasi and Yager [10] investigated an IFS based interpretations of a kind of complex
decision making. Zhang [11] developed some generalized power geometric operators, which are novel tools to
aggregate intuitionistic fuzzy information. While in Ref.
[12-13], two objective decision technologies were put
forward: the TOPSIS method and kinds of objective
ways to obtain the associated weighting vector. It’s
worth noting that the information aggregation processes
and the approaches to get the weights have attracted
many authors’ interests [14-17, 30, 32, 38]. Wu and Cao
[14] proposed several geometric operators to aggregate
intuitionistic trapezoidal fuzzy numbers. Chen [15] proposed the induced generalized continuous ordered
weighted averaging operator and discussed its application. Zhou and Chen [16] developed some generalized
power aggregation operators.
However, although the fuzzy information has been
investigated in decision models and has also been widely
studied, MAGDM is still a difficult process because of
the complexity of real problems. The traditional decision
theories are mainly pay attention on the situation that all
© 2014 TFSA
Z. Tao et al.: A Generalized Multiple Attributes Group Decision Making Approach Based on Intuitionistic Fuzzy Sets
experts are faced with the same attribute set, in other
words, all experts are required to provide their judgments for each alternative under all attributes without
considering whether they are familiar with them or not.
The expert would make mistakes if he/she is enforced to
make judgments that out of his/her capability. Thus, it’s
necessary to consider the practical situation that the attribute information provides for experts is unsymmetrical. An extreme case is, now a company try to purchase
new equipments, five suppliers are selected for further
consideration. As maintenance services is very important
in the process of using, therefore the company need to do
an evaluation about the five suppliers under nine attributes by three departments: the administrative department
consider the customer satisfaction (a1), service attitude
(a2), management level (a3) and company size (a4); the
technology department care about the maintenance of
speed (a5), maintenance of quality (a6), the level of technical advice (a7) and the level of information (a8); the
financial sector have to think about reasonableness of the
charge (a9). Herein, three departments take into account
alternatives via their own attribute information, respectively.
A generalized multiple attributes group decision making model is developed, in which the attributes provided
for experts are not required to be the same. Besides, we
notice that the transformation among types of fuzzy sets
has not been applied in decision theory, thus we propose
an information collection and transformation based
GMAGDM procedure.
We organize the paper as follows: Section 2 briefly
reviews several basic concepts which are needed. Section 3 mainly defines the GMAGDM and gives some
basic description about the new decision model. We
discuss three GMAGDM with different weight information in Section 4, which are associated with completely
known, partly known and completely unknown weighting vector of experts and attributes, respectively. And in
Section 5, we propose a novel information collection and
transformation procedure to deal with GMAGDM with
intuitionistic fuzzy numbers on the basis of hesitant
fuzzy sets. An illustrative example shows the feasibility
and efficiency of our proposed methods in Section 6,
while some conclusions and future work are summarized
and discussed in Section 7.
grees of membership and the degrees of
non-membership of xi to A in X , satisfying the condition that:
0   A  xi   1,0  v A  xi   1,  A  xi   v A  xi   1 ,
(2)
and  A ( xi )  1   A ( xi )  v A ( xi ) is called an hesitant degree
of xi to A .
For convenience, Burillo et al [18] named the pair
 , v an intuitionistic fuzzy number (IFN) denoted as
 with the condition 0   , v  1 ,   v  1 . Then for
any two IFNs    , v
Definition 2.1 [1]: Suppose that X   x1, x2 ,, xn  is the
universe of discourse. An intuitionistic fuzzy set A on X
can be defined as:
(1)
A   xi ,  A  xi  , v A  xi  | xi  X 
where the function  A ( xi ) and v A ( xi ) represent the de-
and     , v
and for any
positive scalar  , the following operational laws are
valid:
(1)              , v  v ;
(2)   1  1    , v .

For any IFN    , v , Chen etc. [19] gave the
concept of score function of  according to,
s ( )    v .
(3)
Then, Hong, etc.[20] proposed the following accuracy
function of  :
h( )    v .
(4)
Then Xu [21] proposed the following method to compare two IFNs on the basis of score function and accuracy function.
Definition 2.2: Assume that 1  1 ,v1 and  2  2 ,v2
are two IFNs,
(1) If s (1 )  s ( 2 ) , then 1 is larger than  2 , denoted by 1   2 .
Otherwise, if s (1 )  s ( 2 ) , and
(2) If h(1 )  h( 2 ) , then 1   2 ;
(3) If h(1 )  h( 2 ) , then 1 is larger than  2 , denoted by 1   2 .
Definition 2.3 [22]: A generalized entropy measure of an
intuitionstic fuzzy set A of n elements is,
E  A 
where
1 n  max Count ( Ai  AiC ) 
.

n i 1  max Count ( Ai  AiC ) 
Ai  AiC  min(Ai , AC ),max(vAi , vAC )
i
i
,
(5)
Ai  AiC 
max(Ai , AC ),min(vAi , vAC ) , AiC  Ai , a point having coori
dinates
2 Preliminaries
185
i
 A , v A , A
C
i
C
i
C
i
 v Ai ,  Ai , Ai
and the notation
n
max Count( A)  (A (xi )   A ( xi ))
represents the biggest
i 1
cardinality of intuitionistic fuzzy set A.
As another extension of traditional fuzzy sets, Torra
etc. [4, 5] developed the notion of hesitant fuzzy set, in
which the membership degrees of an element to a set are
several possible values in the interval of [0 ,1].
International Journal of Fuzzy Systems, Vol. 16, No. 2, June 2014
186
Definition 2.4 [4, 5]: Given that X is a universe of
discourse, a hesitant fuzzy set defined on X is represented as E   x, hE ( x) | x  X  , where hE ( x) is a set of
values in [0, 1] , representing possible membership degrees of the element x  X to the set E. For convenience, hE ( x) is also called a hesitant fuzzy element
(HFE).
Since the precise degrees of membership of an element to a set are often hard to be provided. Chen, Xu
and Xia [23] introduced the case of interval-valued HFSs
(IVHFSs) to overcome the drawback.
Definition 2.5: Given that X is a domain of universe
and D[0, 1] represents the set of all closed subintervals
of [0, 1], an IVHFS on X is
A  xi , hA ( xi ) | xi  X , i  1,2,, n .
(6)


where hA ( xi ) : X  D  0,1 denotes all possible interval
valued degrees of membership of xi  X to A . Similar
to the case of HFS, hA ( xi ) is said to be an interval valued HFE (IVHFE), which can be written as,
(7)
hA ( xi )   |   hA ( xi ) .
Herein,    L , U  is an interval number.  L  inf 
and  U  sup  represent the lower and upper limits of
 , respectively.
Torra and Narukawa [4] and Torra [5] described the
relation between an intuitionistic number and an interval fuzzy set, expressed as follows:
Definition 2.6: Given that A   x,  A ( x), v A ( x)  is an
intuitionistic fuzzy set, the definition of the corresponding hesitant fuzzy set is straightforward:
(8)
h( x )    A ( x ),1  v A ( x )  if  A ( x)  1  v A ( x) .
For extending the length of IVHFEs and comparing
any two IVHFEs, the following two definitions are valid.
Definition 2.7 [24]: Suppose that a   a  , a   and
b  b  , b   are any two interval-valued fuzzy numbers
(IFNs), and let la  a   a  and lb  b   b  , the degree of
possibility of a  b is given by



 b  a  
(9)
,0  ,0  .
p a  b  max 1  max 
 la  lb  

Definition 2.8 [23]: For an IVHFE h , the score function


1
  , where lh denotes the
lh  h
number of the interval values of h .
If s(h )  s(h ) , then h  h .
of h is defined by s (h ) 
1
2
1
2
3. The Concept of Generalized Multi-Attributes
Decision Making
Next, we give the definition of generalized
multi-attributes decision making in the following:
Definition 3.1: Let X be the set of m alternatives and E
be the set of l experts, given Ak  A (A is the attribute set
with n attributes) be the corresponding attribute set of
the k-th expert ek , where lk 1 Ak  A and the subsets Ak
(k=1, 2,  , l) are not required to be the same. Therefore, a generalized multiple attributes group decision
making (GMAGDM) is aggregating the decision information provided by experts for all alternatives under
their own attribute set Ak (k=1, 2, …, l).
There are some interesting relations among MADM,
MAGDM and GMAGDM problems, which can be listed
in the following:
Remark 1: The GMAGDM degenerates to be a MADM
if for any Ap and Aq, we have Ap  Aq=  , where
p  q, p, q  1,2,, l .
In this case, each attribute is evaluated for all alternatives by all experts for only one time. It would be the
same as the situation in MADM while considering the
weights of experts.
Remark 2: The GMAGDM degenerates to be a
MAGDM if for any Ap and Aq, we have Ap  Aq , where
p  q, p, q  1,2,, l .
Actually, Ap  Aq means that the attributes provided
to each decision maker are the same, which is consistent
with MAGDM problem.
Remark 3: Definition 3.1 shows that the MADM and
MAGDM are two special situations of the GMAGDM.
In other words, the GMAGDM is the generalized form
of multiple attributes group decision making problems.
According to the relations among MADM, MAGDM
and GMAGDM problems, it’s easy to find that
GMAGDM summarizes the existing types of decision
making problems, which is the most general decision
making process.
Therefore, the solving of GMAGDM is meaningful
for decision making theory. In the intuitionistic fuzzy
environment, a generalized multi-attributes group decision making issue is composed of the following elements.
Assume that X={x1, x2,  , xm} is a finite set of alternatives and E={e1, e2,  , el} is a finite set of experts with
associated weighting vector W={w1, w2, …, wl}, and
A={a1, a2,  , an} is a group of n attributes with
weighting vector   1 , 2 ,, n  , where some of its
subsets, denoted as Ak (k=1, 2,  , l), are considered by
the k-th expert. Each expert ek  E presents his/her
evaluation on xi under Ak, and the evaluations are pro-
Z. Tao et al.: A Generalized Multiple Attributes Group Decision Making Approach Based on Intuitionistic Fuzzy Sets
vided with IFNs, denoted as rij k   ij k  , vij k  , where
i  1, 2,  , m ; j  J k ; k  1, 2,  , l , herein, J k is the
subscript set of attributes in Ak, the decision information
given by the k-th expert can be denoted as Rk  (rij k  ) m A
k
k 
( Ak represents the potency of Ak ), ij
k 
and vij
sat-
isfy the condition that ij 0,1, vij  0,1, ij  vij  1 ,
which represent the satisfactory degree and unsatisfactory degree , respectively.
Thus, a GMAGDM problem is to obtain a final ranking for alternatives through certain decision procedures.
k
k
k
k 
4. Information Aggregation Based GMAGDM
Models
Herrera, Martinez and Sanchez [25] summarized a
general procedure for group decision making with fuzzy
multiple attributes problem into three steps: The first
step is standardizing the evaluation values given by experts (except that the evaluations are provided in the
form of preference information). The second step is obtaining a collective opinion for each alternative using
both direct and indirect aggregation approaches. The final step would select preferred alternative(s) on the basis
of ranking order. Therefore, the second stage, in which
the decision information given by each expert is aggregated and a MAGDM problem would then be transformed into a MADM issue, turns to be a process of
great concern.
Herein, motivated by the work of Herrera, Martinez
and Sanchez [25] we are going to propose some general
procedures to handle the information of a GMAGDM
problem.
Let Vi k  (rij k )1 A and Vj k   (r1jk  , r2 kj  ,, rmj k  )T be the colk
lection of the evaluated values for the i-th alternative
under all attributes and the collection of evaluated values
for all alternatives under the j-th attribute given by the
k-th expert, respectively, where i =1, 2, , m, k =1, 2, , l ;
j  Jk .
An n-dimensional function f : V n  V is called an
n-ary aggregation function if it is monotone
non-decreasing in each place and fulfills the following
boundary conditions:
inf f (v )  inf V and sup f (v)  sup V .
vV n
vV n
Hence, information aggregation process for each expert would be synthesizing the values Vi ( k ) or V j( k ) using an n-ary aggregation function with associated
weighting information.
With these notations, we now show several models to
handle intuitionistic fuzzy sets based generalized multi-
187
ple attributes group decision making under diverse
weight information’s environments.
Case 1. GMAGDM with completely known weight information
If a decision maker know the importance of all experts
and attributes and provide the weight information of
them beforehand, we say this situation as the weight information is completely known, we can give an algorithm as follows:
Algorithm 1:
Step 1. Input the initial evaluated information

Rk  ( rij k  ) m A ;
k
Step 2. Compute the synthetical evaluation values of
each alternative given by experts using the weight information of attribute, denoted as
ri ( k )  f (Vi  k  ), i  1,2,, m; j  J k ,
(10)
(k )
where ri represents the synthetical evaluation values
given by the k-th experts.
Step 3. Compute the final synthetical evaluation values of each alternative using the weight information of
experts, denoted as ri (i  1,2,,9) , where
ri  f (ri 1 , ri  2 ,, ri  l  ) .
(11)
Step 4. Rank all alternatives according to corresponding sore and accuracy values and select the optimal decision according to Definition 2.2.
Remark 4: The aggregation function plays an important
role in the decision process, and different aggregation
functions may produce diverse results. Many aggregation functions for intuitionistic fuzzy values have been
proposed: some averaging operators [26], the induced
generalized intuitionistic fuzzy functions [27] and OWA
operator [28], the induced correlated aggregation operators [29], the intuitionistic fuzzy power geometric operators [11], etc. As a result, equations (10) and (11)
could be replaced by various information aggregation
operators. Hereinafter, we will choose the weighted averaging aggregation operator in our methodology and
compare the results obtained by different approaches.
Case 2. GMAGDM with partial weight information
If the decision maker owns certain preference to experts and attributes and give some weight information,
but not exactly, denoted as W  ,    , where  and
 represent the associated weighting information of
the attributes and experts, respectively. Normally, they
have the following 6 situations [43-44]: i   j ,
i   j   i , i  i  j ,  i  i   i   i , herein,  i , i ,  i , i
and i are all non-negative numbers and  is the corresponding weights. Then we call this situation as the
weight information is partly known.
Chen and Yang [30] proposed a novel approach for
MAGDM, in which the evaluation values for all alterna-
International Journal of Fuzzy Systems, Vol. 16, No. 2, June 2014
188
tives under the same attribute given by different experts
are aggregated firstly, and then aggregate final decision
values by taking into account the attribute weights. In
Wei [31], the maximizing deviation theory was applied
in MADM with intuitionistic fuzzy sets. Herein, motivated by these contributions, we introduce an extension
approach based on the maximizing deviation theory for
GMAGDM with partly known weight information. And
the corresponding algorithm can be given.
Algorithm 2:
Step 1. Input the initial evaluated decision values

Rk  ( rij k  ) m A ;
k
Step 2. Compute the associated weighting vector of
experts W={w1, w2, …, wl} using the method of maximizing deviations theory, which can be written by the
following optimization model,
m n
n
l

 l
k 
k  
 max f W    d   wk rij ,  wk rij  

i 1 j 1 j  1  k 1
k 1
.
 s.t. W  

(12)
l
where rij   wk rij( k ) represents the synthetical IFN of the
k 1
i-th alternative under the j-th attribute given by all experts; function d ( ,  ) is utilized to measure the difference between two IFNs  and  .
Optimal model (12) is an optimization problem. We
can solve the model (12) using some algorithms, for instance, feasible direction method, constraint function
method (exterior point method, EPM).
Step 3. Aggregate the decision information under the
j-th attribute and obtain attribute synthetical values, denoted as rj  (rij )m1 .
Thus, the initial GMAGDM is transformed into a
MADM problem.
Step 4. Repeat Step 2 for the aggregation values of all
attributes, and get the weights of all attributes
  1 , 2 ,, n  , which can be shown as follows:
m
n

 n

 max f W    d    j rij ,   j rij  



1
1
i 1
j
j

.

 s.t.   
(13)
Step 5. Aggregate the final decision values of alternatives r  (ri )m1 on the basis of attributes’ associated
weighting vector   1, 2 ,, n  obtained in Step 4
and attribute synthetical values obtained in Step 3, using
the aggregation function,
ri  f (ri1 , ri 2 ,, rin ) .
Step 6. Rank all alternatives according to corresponding sore and accuracy values and select the optimal decision according to Definition 2.2.
Noted also that the measure in objective function
plays an important role in the whole process, distance is
an normal and widely used measure in the maximum
deviation theory, also many intuitionistic distance measures have been proposed [33, 34]. Besides, there are
kinds of indices for measuring the difference between
IFNs, such as similarity measures [35, 36], cross-entropy
measure [37], thus model (12) and (13) could be represented in diverse formats.
Case 3. GMAGDM with completely unknown weight information
In practical problems, because of kinds of uncertainty,
it’s hard to determine or give a right evaluation for the
weight information. Thus, we say this situation as the
weight information is completely unknown. Similar to
case 1, we construct an algorithm including two more
steps to determine the weights.
Algorithm 3:
Step 1. Input the initial evaluated decision values

Rk  ( rij k  ) m A ;
k
Step 2. Compute the weight of the j-th attribute of
Ak , j  J k using entropy of V j k  according to Eq. (5),
k
noted as H (V j  ) , where j  J k .
The weight of the j-th attribute in Ak , j  J k noted as
 can be calculated by
k
j
 jk 
1  H (V j k  )
Ak 
 H (V   )
k
j J k
.
(14)
j
The weight gotten by Eq. (14) has some advantages:
the attributes’ weights are varied for different expert; the
closer among evaluation values under an attribute, the
less the weight is. This can reduce the influence among
experts and avoid the information redundancy.
Step 3. Aggregate the evaluated values given by the
k-th expert according to Eq. (10).
Then the decision matrix with synthetical evaluated values of each alternative can be obtained, and thus the
GMAGDM problem is transformed into a MADM
model.
Step 4. Repeat Step 2 for the aggregation values of all
experts, and get the weight of the k-th expert, denoted as
wk , which can be counted as,
wk 
1  H (rk )
l
l   H (rt )
.
(15)
t 1
Step 5. Aggregate the final decision values of alternatives on the basis of formula (11) using the associated
weight vector of experts W.
Step 6. Rank all alternatives according to corresponding sore and accuracy values and select the optimal decision according to Definition 2.2.
The differences between Algorithm 2 and Algorithm 3
are the order of information aggregation and ways to
Z. Tao et al.: A Generalized Multiple Attributes Group Decision Making Approach Based on Intuitionistic Fuzzy Sets
obtain the weighting vectors of experts and attributes.
Since there would be information loss in the aggregation
process, as a result, hereinafter, we are going to propose
a novel approach on the basis of information transformation.
5. Information Transformation Based
GMAGDM Approach



189

1
d3  ,   max i L(i )  L(i )  U(i )  U(i ) ,
2
and



2
2
1
d4  ,   max i L(i )  L(i )  U(i )  U(i )
2

(18)
12
,
(19)
Similarly, we can deduce two corresponding hybrid
distance measures for IVHFEs by combining formulas
(16-19), shown as


1 1
d5  ,     Li   Li   U i   U i  
4  l i 1
,
L
L
U
U



max i  i     i     i     i  

l
2
2
1 1
d6  ,   
  L   L  Ui  Ui 
2  2l i 1   i   i 
2
2 1 2
1
maxi L i   Li   U i   Ui 

2



l
According to the definition of hesitant fuzzy sets, an
(20)
HFS can be seen as a collection of possible memberships.
While in a GMAGDM model, the evaluations of one
alternative under a fixed attribute are provided by some
 
experts, but may not be all of them. Therefore, we col. (21)
lect these values and see them as possible memberships.
An information collection and transformation process
can be described as follows:
Eq. (18) to Eq. (20) have the properties listed in the
Definition 5.1: Suppose that Vij  {rij k  , j  J k ; k  1, 2,, l}
following:
is the collection of the evaluated values for the i-th al(1) 0  d ( ,  )  1 ;
ternative under the j-th attribute provided by the all ex(2) d ( ,  )  0 if and only if    ;
perts. The corresponding transformed hesitant fuzzy
(3) d ( ,  )  d (  , ) .
element of Vij , denoted by hij , can be defined by
In view of above analysis and definitions, now we
hij   ij( k ) ,1  vij( k )  , k  1,2,, l , if ij( k )  1  vij( k ) , i=1, 2, …,
show an information collection and transformation prom; j  J k ;
cedure to GMAGDM:
It’s worth noting that if for all i  1, 2, , m ; j  J k ; Algorithm 4:
Step 1. Input the initial evaluated decision information
k  1, 2, , l , ij( k )  1  vij( k ) , then hij degenerates to be a
k 

classical HFE, while in the interval valued hesitant fuzzy Rk  (rij )m A ;
Step 2. Collect the evaluation values of all alternatives
environment
we
still
use
the
notation
h    ( k ) ,1  v( k )  , k  1,2,, l .
under
the same attribute given by different experts, deij
ij 
 ij
noted as hij  {rij k } .
Suppose that  and  be any two IVHFEs with
Step 3. Transform hij into interval hesitant fuzzy
the same length l and the elements in  and  are
both arranged in a descending order. In Ref. [34], the element, denoted as ij , where
Hamming distance and Euclidean distance for IVHFEs
ij   ij( k ) ,1  vij( k )  , if ij( k )  1  vij( k ) and rij( k )  rij( k )
were proposed, which can be shown as follows:
( k  k , k , k   1,2,, l ).
1 l
(16)
d1  ,     Li   Li   Ui   U i  .
Then i  i1 , i 2 ,, in  represents the hesitant fuzzy
2l i 1
l
2
2
decision information of the i-th alternative.
1
(17)
d 2  ,   
  L  Li   Ui  Ui .
Thus, the initial GMAGDM with intuitionistic fuzzy
2l i 1   i 
decision
information is transformed into a MADM
where  (i ) and  (i ) be the i-th largest value in 
problem under interval-valued hesitant fuzzy environand  .
ment.
Drawing on the hesitant distances introduced in Ref.
Step 4. Extend the interval hesitant fuzzy sets with
[23] and the concepts of some extended distance meas- lower dimension with optimistic so that the hesitant
ures defined in Ref. [33-34, 38-39], the following defini- fuzzy decision information under the same attribute own
tions are considered:
identical dimension according to Definition 2.7, 2.8.
Definition 5.2: Given any two IVHFEs  and  with
Step 5. Choose the positive and negative ideal
the same length l and the elements are both arranged in a IVHFSs for all alternatives under each attribute using
descending order. The following two extended distance comparing law of IVHFSs defined in Definition 2.8, demeasures between IVHFEs based on the Hausdorff dis- noted as    1 , 2 ,, n  and    1 , 2 ,, n  , retance are valid:
spectively.




k






International Journal of Fuzzy Systems, Vol. 16, No. 2, June 2014
190
Step 6. Calculate the distance d  (i ,   ) and d  (i ,   )
based on formulas (16-21), respectively.
Step 7. Compute the relative closeness coefficient
c (i ) of the i-th alternative xi , shown as follows:
d  (i ,   )
c(i )  
.
d (i ,   )  d  (i ,   )
Step 8. Rank the values of c(i ) in a descending sequence on the basis of Definition 2.8, where i=1, 2,  ,
m. The larger the value of c(i ) , the more closer between xi and positive ideal IVHFSs   , and the more
removed between xi and negative ideal IVHFSs   .
Thus, the alternative with the maximum relative closeness coefficient should be selected as the best choice.
Remark 5: Algorithm 3 provides a novel way to deal
with GMAGDM issue, which can be easily applied and
extended to normal MAGDM problems. In the whole
process, the requirement of weights of experts and attributes are avoided, it is simple and information
well-utilized.
6. Illustrative Example
In this section, a numerical example is developed to
show the application of the new approaches. An investment selection problem adapted from Zeng and Su [38]
is used to look for an optimal investment.
A decision maker would like to have investment in
one company, six possible companies are considered to
be invested according to his/her market research, which
are a chemical company (x1),a food company (x2), a
computer company (x3),a car company (x4), a furniture
company ( x5 ) and a pharmaceutical company ( x6 ).
The investor has brought together a set of experts to
evaluate these alternatives. The set of experts is constituted by three persons denoted as E={e1, e2, e3}. They use
six attributes to evaluate the ability of companies, i.e.,
short-term yield (a1), medium-term yield (a2), long-term
yield (a3), risk (a4), investment difficulty (a5) and other
factors (a6).
Three experts’ corresponding attribute sets are A1={a1,
a2, a3, a6}, A2={a4, a5, a6} and A3={a1, a2, a3, a4, a5, a6}.
Experts provide their opinions regarding the results
corresponding to each company, and the results are listed
in Tables 1, 2, 3, which are represented in IFNs.
Drawing on the decision information, we rank the
companies by the proposed decision making method.
Since case 3 is an extension of case 1, herein we just
shown the latter three cases, and assume that in case 2,
the weight information  be given as,


 W   w1, w2, w3  | 0.2  w1  0.6,0.3  w2  0.5,0.1 w1  0.4,wi 1 ,
i

1


3
   1, 2,, 6  | 0.1  1  0.2,0.18  2  0.21,0.15  3  0.22,


6
  
.
0.09
0.25,0.12
0.14,0.01
0.11,
1












4
5
6
i


i 1


Considering the weighted arithmetic averaging (WAA)
operator as an example, hereinafter, we provide three
GMAGDM processes using Algorithm 2-4.
Algorithm 2:
Step 1. Input the initial intuitionistic fuzzy decision
information list in Table 1-3, denoted as R1  (rij1 )6 4 ,
R 2  (rij 2 )63 , R3  ( rij3 )66 ;
Step 2. Construct optimization model (10) using the
Hamming distance as the tool to measure the difference
among IFNs, the preference information of the decision
maker is given as assumed  . Then we obtain the
weights of experts W   0.2,0.5,0.3 .
Step 3. Aggregate the attribute aggregation values using W, and the results are given in Table 4.
Step 4. Construct optimization model (11), the preference information of the attribute is given as assumed
 . Then we get the weights of attributes, which can be
listed as follows:
   0.2,0.21,0.22,0.12,0.14,0.11 .
Table 1. Evaluated values given by e1.
a1
a2
a3
a6
x1
0.5,0.4
0.5,0.3
0.2,0.6
0.3,0.5
x2
0.7,0.3
0.7,0.3
0.6,0.2
0.4,0.5
x3
0.5,0.4
0.6,0.4
0.6,0.2
0.4,0.4
x4
0.7,0.2
0.7,0.2
0.4,0.2
0.6,0.3
x5
0.4,0.3
0.5,0.2
0.4,0.5
0.7,0.2
x6
0.6,0.2
0.4,0.3
0.7,0.3
0.6,0.2
Table 2. Evaluated values given by e2.
a4
a5
a6
x1
0.7,0.2
0.6,0.3
0.5,0.4
x2
0.3,0.4
0.7,0.1
0.8,0.2
x3
0.7,0.1
0.3,0.4
0.6,0.3
x4
0.6,0.2
0.4,0.5
0.5,0.2
x5
0.4,0.3
0.7,0.2
0.4,0.3
x6
0.6,0.2
0.5,0.3
0.7,0.2
Z. Tao et al.: A Generalized Multiple Attributes Group Decision Making Approach Based on Intuitionistic Fuzzy Sets
191
Table 3. Evaluated values given by e3.
a1
a2
a3
a4
a5
a6
x1
0.5,0.3
0.7,0.2
0.5,0.3
0.5,0.4
0.7,0.3
0.4,0.3
x2
0.6,0.3
0.6,0.2
0.7,0.2
0.8,0.1
0.5,0.4
0.6,0.2
x3
0.7,0.3
0.4,0.4
0.6,0.3
0.4,0.2
0.6,0.3
0.4,0.4
x4
0.4,0.4
0.6,0.2
0.4,0.2
0.7,0.2
0.6,0.2
0.5,0.3
x5
0.7,0.2
0.7,0.3
0.6,0.1
0.7,0.3
0.5,0.3
0.3,0.4
x6
0.5,0.2
0.5,0.3
0.8,0.2
0.6,0.1
0.6,0.2
0.6,0.2
Table 4. Aggregation attributes values of alternatives.
a1
a2
a3
a4
a5
a6
x1
0.2929,0.5802
0.3934,0.4850
0.2232,0.6292
0.5551,0.3397
0.5593,0.3817
0.4351,0.3837
x2
0.4029,0.5477
0.4029,0.4850
0.4198,0.4472
0.4838,0.3170
0.5551,0.2402
0.6933,0.2297
x3
0.3934,0.5802
0.2857,0.6325
0.3675,0.5051
0.5301,0.1951
0.3644,0.4407
0.5101,0.3464
x4
0.3257,0.5506
0.4029,0.4472
0.2254,0.4472
0.5593,0.2759
0.4116,0.4363
0.5218,0.2449
x5
0.3708,0.4850
0.3934,0.5051
0.3141,0.4363
0.4602,0.3817
0.5551,0.3116
0.4529,0.3016
x6
0.3238,0.4472
0.2666,0.5477
0.5150,0.4850
0.5196,0.2241
0.4628,0.3380
0.6536,0.2000
Step 5. Aggregate the final decision values of all alternatives, the aggregation values are displayed in the
following:
r1  0.3962,0.4807 , r2  0.4800,0.3872 , r3  0.3959,0.4571 ,
r4  0.3917,0.4104 , r5  0.4139,0.4142 , r6  0.4480,0.3848 .
Step 6. The scores of above final decision values are,
s  r1   0.0844, s  r2   0.0928, s  r3   0.0612 ;
s  r4   0.0187, s  r5   0.0003, s  r6   0.0631 .
Then, according to the scores, we can deduce that x2
is the best alternative according to Definition 2.2.
The solution produced by Algorithm 3 can be listed as
follows:
Algorithm 3:
Step 1. Input the initial intuitionistic fuzzy decision
information list in Table 1-3, denoted as R1  (rij1 )6 4 ,
R 2  (rij 2 )63 , R3  ( rij3 )66 ;
Step 2. Compute the entropy of evaluation values under each attribute provided by each expert, the results are
listed in the following Table 5.
Then according to the definition of entropy weight,
three associated weighting vector of attributes can be
calculated and listed by,
1   0.2414,0.2592,0.2765,0,0,0.2229  ;
 2   0,0,0,0.3492,0.3137,0.3371 ;
 3   0.1562,0.1696,0.1978,0.2014,0.1676,0.1074  .
Step 3. Aggregate the evaluated values given by three
experts and the results are listed in Table 6.
Table 5. Entropy of evaluation values under each attribute
provided by experts.
a1
a2
a3
a4
a5
a6
e1
0.6379
0.6111
0.5853
1
1
0.6657
e2
1
1
1
0.4296
0.3137
0.3371
e3
0.6190
0.5863
0.5175
0.5089
0.5913
0.7381
Table 6. Aggregating experts’ evaluation values.
e1
e2
e3
x1
0.3863,0.4365
0.6070,0.2869
0.5708,0.2968
x2
0.6209,0.3005
0.6482,0.2050
0.6588,0.2081
x3
0.5379,0.3302
0.5688,0.2237
0.5357,0.2994
x4
0.6126,0.2189
0.5103,0.2666
0.5537,0.2328
x5
0.5096,0.2842
0.5173,0.2642
0.6211,0.2337
x6
0.5896,0.2485
0.6106,0.2271
0.6250,0.1863
192





h 






International Journal of Fuzzy Systems, Vol. 16, No. 2, June 2014
0.5,0.4 , 0.5,0.3 , 0.5,0.3 , 0.7,0.2 , 0.2,0.6 , 0.5,0.3 , 0.7,0.2 , 0.5,0.4 , 0.6,0.3 , 0.7,0.3 , 0.3,0.5 , 0.5,0.4 , 0.4,0.3  

0.7,0.3 , 0.6,0.3 , 0.7,0.3 , 0.6,0.2 , 0.6,0.2 , 0.7,0.2 , 0.3,0.4 , 0.8,0.1  , 0.7,0.1 , 0.5,0.4 , 0.4,0.5 , 0.8,0.2 , 0.6,0.2  

0.5,0.4 , 0.7,0.3 , 0.6,0.4 , 0.4,0.4 , 0.6,0.2 , 0.6,0.3 , 0.7,0.1 , 0.4,0.2 , 0.3,0.4 , 0.6,0.3 , 0.4,0.4 , 0.6,0.3 , 0.4,0.4  

0.7,0.2 , 0.4,0.4 , 0.7,0.2 , 0.6,0.2  , 0.4,0.2 , 0.4,0.2 , 0.6,0.2 , 0.7,0.2 , 0.4,0.5 , 0.6,0.2 , 0.6,0.3 , 0.5,0.2 , 0.5,0.3  

0.4,0.3 , 0.7,0.2 , 0.5,0.2 , 0.7,0.3 , 0.4,0.5 , 0.6,0.1 , 0.4,0.3 , 0.7,0.3 , 0.7,0.2 , 0.5,0.3 , 0.7,0.2 , 0.4,0.3 , 0.3,0.4  

0.6,0.2 , 0.5,0.2 , 0.4,0.3 , 0.5,0.3 , 0.7,0.3 , 0.8,0.2 , 0.6,0.2 , 0.6,0.1  , 0.5,0.3 , 0.6,0.2  , 0.6,0.2 , 0.7,0.2 , 0.6,0.2  
(22)
 0.5,0.7, 0.5,0.6 , 0.7,0.9 , 0.5,0.7 , 0.5,0.7, 0.2,0.4 , 0.7,0.8 , 0.5,0.6 , 0.7,0.7 , 0.6,0.7 , 0.4,0.7 , 0.5,0.6, 0.3,0.5 


 0.7,0.7, 0.6,0.7 , 0.6,0.8 , 0.7,0.7 , 0.7,0.8, 0.6,0.8 , 0.8,0.9 , 0.3,0.6 , 0.7,0.9 , 0.5,0.6 , 0.8,0.8, 0.6,0.8 , 0.4,0.5 


 0.7,0.7, 0.5,0.6 , 0.6,0.6, 0.4,0.6 , 0.6,0.8, 0.6,0.7 , 0.7,0.9 , 0.4,0.8 , 0.6,0.7 , 0.3,0.6 , 0.6,0.7 , 0.6,0.7, 0.4,0.6 

  
 0.7,0.8, 0.4,0.6 , 0.7,0.8 , 0.6,0.8 , 0.4,0.8 , 0.4,0.8 , 0.7,0.9 , 0.6,0.8 , 0.6,0.8, 0.4,0.5 , 0.5,0.8 , 0.6,0.7 , 0.5,0.7 


 0.7,0.8, 0.4,0.7 , 0.7,0.7 , 0.5,0.7 , 0.6,0.9 , 0.4,0.5 , 0.7,0.7 , 0.4,0.7 , 0.7,0.9 , 0.5,0.7 , 0.7,0.8 , 0.4,0.7 , 0.3,0.6 


 0.6,0.8 , 0.5,0.7 , 0.5,0.8 , 0.4,0.7 , 0.8,0.8 , 0.7,0.7 , 0.6,0.9 , 0.6,0.8 , 0.6,0.8 , 0.5,0.8 , 0.7,0.8, 0.7,0.8, 0.6,0.8 


(23)
Step 4. Repeat Step 2 for the aggregation values of all
experts, and get the experts’ weight
W   0.2827,0.3499,0.3674  .
Step 5. Aggregate the final decision values of alternatives on the basis of experts’ associated weighting vector
W, and the final evaluation values of alternatives are,
r1  0.5408,0.3271 , r2  0.6447,0.2297 , r3  0.5482,0.2780 ,
r4  0.5570,0.2399 , r5  0.5564,0.2578 , r6  0.6102,0.2166 .
Step 6. The scores of above final decision values are,
s  r1   0.2137, s  r2   0.4150, s  r3   0.2702 ;
s  r4   0.3171, s  r5   0.2986, s  r6   0.3936 .
Then, according to the scores, we can deduce that x2
is the best alternative, which is the same as Algorithm 2.
And the solution given by Algorithm 4 can be shown
in next steps:
Step 1. Input the initial intuitionistic fuzzy decision
information in Table 1-3, denoted as R1  (rij1 )6 4 ,
R 2  (rij 2 )63 , R3  ( rij3 )66 ;
Step 2. Collect the evaluations of all alternatives under
the same attribute given by different experts, then we
have Eq. (22), where the element in the matrix represents
the collections of the i-th alternative under the j-th attribute given by different experts, denoted as hij  rij k   .
Step 3. Transform hij into interval hesitant fuzzy
element, denoted as ij , herein Eq. (23).
Step 4. Extend the interval hesitant fuzzy sets with
lower dimension so that the hesitant fuzzy decision information under the same attribute own identical dimension (noted that we finish this step in Step 3 and the results are shown also in  ).
Step 5. Choose the positive ideal IVHFSs and negative ideal IVHFSs for all alternatives under each attrib-
ute using the comparing law of IVHFSs defined in Definition 2.7, 2.8, where
  0.7,0.7,0.6,0.7 , 0.7,0.8, 0.6,0.8 ,0.8,0.8,0.7,0.7 ,
0.7,0.9,0.6,0.8,0.7,0.9,0.5,0.7,0.4,0.7,0.5,0.6,0.3,0.5
 0.5,0.7,0.5,0.6 , 0.6,0.6,0.4,0.6 ,0.5,0.7, 0.2,0.4 ,
.
0.7,0.7,0.4,0.7,0.6,0.7,0.3,0.6,0.7,0.8,0.7,0.8,0.6,0.8
,

Step 6. Compute the distance between i and two
ideal HFSs using formula (16), where
d 1,   0.1083,d  1,   0.0889,d  2,   0.0792,d 2,   0.1403 ;
d 3,   0.1000,d 3,   0.0889,d 4,   0.0931,d 4,   0.1208 ;
d 5,   0.0986,d 5,   0.1125,d 6,   0.0889,d 6,   0.1167 .
Step 7. Compute the relative closeness coefficient
xi .
c  1   0.4507, c  2   0.6392, c  3   0.4706 ;
c  4   0.5649, c  5   0.5329, c  6   0.5676 .
Step 8. Rank the values of c(i ) in a descending sequence, thus the best alternative should be x2 , which is
also equivalent to the choice that obtained in former
procedures.
The relative closeness coefficients caused by other
distance measures can be listed in the following Table 7.
From Table 7, the food company ( x2 ) is always the
best alternative, which means that the results deduced by
different distance measures are consistent.
c (i ) of the alternative
7 Conclusions
This paper initiates a concept of generalized multiple
attribute group decision making, which will bring much
applicability for our researches on group decision making theory. Based on this notion, some relevant situations
Z. Tao et al.: A Generalized Multiple Attributes Group Decision Making Approach Based on Intuitionistic Fuzzy Sets
with diverse weight information have been discussed.
We introduce three general procedures to handle corresponding different weight information based GMAGDM
problems. Although many authors had studied decision
making issues on the basis of intuitionistic, they mainly
consider various approaches to produce the associated
weighting vectors of experts and attributes in decision
process, including some subjective and objective ways,
while in this paper, we propose an information collection
and transformation procedure to deal with such kind of
decision problem. We find out that it is a direct and objective approach for the solution of GMAGDM, and it
can also be applied in normal group decision process.
Finally, we give an illustrative example to show the reasonability and efficiency of these new methods proposed
in our paper.
Since we also take notice of the situation that the attribute information provided for experts in a GMAGDM
problem can be more easily expressed in the form of soft
sets [40] or kinds of fuzzy soft sets theory [41-42]. Thus,
it would be an interesting topic to consider the application of soft sets theory in GMAGDM.
Table 7. Relative closeness coefficients obtained by other distance measures.
d 2 ( ,  )
d 3 ( ,  )
d 4 ( ,  )
d 5 ( ,  )
d 6 ( ,  )
c(1 )
0.4353
0.4516
0.4315
0.4512
0.4335
c(2 )
0.6284
0.6250
0.6205
0.6307
0.6246
c(3 )
0.4919
0.5313
0.5257
0.5061
0.5080
c(4 )
0.5688
0.5882
0.5734
0.5782
0.5710
c(5 )
0.5126
0.5000
0.4857
0.5145
0.5004
c(6 )
0.5632
0.5625
0.5636
0.5647
0.5634
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Acknowledgments
This work was supported by Natural Science Foundation of China (71371011, 71301001), Higher School
Specialized Research Fund for the Doctoral Program
(20123401110001), Provincial Natural Science Research
Project of Anhui Colleges (KJ2012A026), The Scientific
Research Foundation of the Returned Overseas Chinese
Scholars, Anhui Provincial Natural Science Foundation
(1308085QG127), Humanities and Social Science Research Project of Department of Education of Anhui
Province (SK2013B041).
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Zhifu Tao is a Ph.D. candidate of Statistics in the School of Mathematical Sciences at the Anhui University. He has
contributed over 10 journal articles to
professional journals. His current research interests include decision making
theory, forecasting, information fusion,
fuzzy statistics and fuzzy mathematics.
Huayou Chen is a Professor of School
of Mathematical Sciences, Anhui University, China. He received a Ph.D. degree in Operational Research from University of Science Technology of China
in 2002. He graduated from Nanjing
University for 2 years postdoctoral research work in 2005. He has published a
book: The Efficient Theory of Combined
Forecasting and Applications(Science Press, Beijing,2008)
and has contributed over 120 journal articles to professional
journals, such as Fuzzy Sets and Systems, Information Sciences, Group Decision and Negotiation etc. His current research interests include information fusion, multi-criteria decision making, aggregation operators and combined forecasting.
Ligang Zhou is an associate professor of
School of Mathematical Sciences, Anhui
University. He received a PhD degree in
operations research from Anhui University in 2013. He has contributed over 40
journal articles to professional journals,
such as Fuzzy Sets and Systems, Applied
Mathematical Modelling, Applied Soft
Computing, Group Decision and Negotiation, Expert Systems with Applications etc. His current research interests include group decision making, aggregation
operators and combined forecasting.
Jinpei Liu is a lecturer of School of
Business, Anhui University, China. He
received a Ph.D. degree in management
science and engineering from Tianjin
University in 2012. He has contributed
over 20 journal articles to professional
journals. His current research interests
include information fusion and combined
forecasting.
195