J. Serv. Sci. & Management. 2008, 1: 11-20
Published Online June 2008 in SciRes (www.SRPublishing.org/journal/jssm)
Extended TOPSISs for Belief Group Decision Making
Chao Fu
School of Management, Hefei University of Technology, Hefei 230009, China
ABSTRACT
Multiple attribute decision analysis (MADA) problems in the situation of belief group decision making (BGDM) are a
special class of decision problems, where the attribute evaluations of each decision maker (DM) are represented by
belief functions. In order to solve these special problems, in this paper, TOPSIS (technique for order preference by
similarity to ideal solution) model is extended by three approaches, by which group preferences are aggregated in dif-
ferent manners. Corresponding to the three approaches, three extended TOPSIS models, the pre-model, post-model,
and inter-model, are developed and their procedures are elaborated step by step. Aggregating group preferences in the
three extended models respectively depends on Dempster’s rule or its modifications, some social choice functions, and
some mean approaches. Furthermore, a numerical example clearly illustrates the procedures of the three extended
models for BGDM.
Keywords: basic belief assignment, belief group decision making, belief preferences aggregation, TOPSIS
1. Introduction
Recently, the uncertain multiple attribute decision analy-
sis (MADA) problems with a group of decision makers
(DMs) have been widely studied in the literature, in which
the attribute evaluations are unknown, vague, partial
known, or imprecise. The representative solution is to
construct a fuzzy TOPSIS (technique for order preference
by similarity to ideal solution), a classical modified ap-
proach for uncertain MADA problems, to choose the best
one from a set of alternatives [2-4, 18, 20, 30].
However, compared with the Dempster-Shafer theory
(DST) [5,23], the operators of fuzzy set theory (FST) to
aggregate group preferences, which are usually the ar-
ithmetical mean, the geometric mean, or their modifica-
tions, are less adaptable and available. Hence, this paper
uses the DST to describe uncertain MADA problems; that
is to say, it uses basic belief assignments (bbas) to repre-
sent uncertain attribute evaluations.
In practice, due to the one-to-one correspondence be-
tween the bba and the belief function [23], the bba is
usually either elicited from experts, or constructed from
observation data. To transform qualitative experts’ opin-
ions into bbas, some methods have been proposed by
Wong and Lingras [31], Bryson and Mobolurin [1], and
Yaghlane et al. [34]. Using the bba to represent uncertain
group attribute evaluations, one correspondingly converts
the group decision making (GDM) to the belief group
decision making (BGDM).
To solve MADA problems in the situation of BGDM,
the original TOPSIS [15] is extended by three approaches
described in [25]. Their operators to aggregate group
preferences are respectively the pre-operation,
post-opera- tion, and inter-operation.
Based on Yang’s rule and utility based equivalent
transformation of the assessments on different frames of
discernment [35], the evaluations on different attributes
related to different frames can be unified to become the
ones on a common frame. Furthermore, the positive and
negative preference vectors of DM, the positive ideal
solution of belief (PISB), and the negative ideal solution
of belief (NISB) are constructed. The preference vectors
avoid the possible paradoxes between the calculating
ranks of alternatives and the fact of DM’s preference, and
the PISB and NISB are used to determine the ranks of
alternatives. The detailed extended models are explained
step by step in Section 3.
The rest of this paper is organized as follows. In Section
2, the related foundations are reviewed. Section 3 dis-
cusses three extended models in accord with three ap-
proaches to aggregating group preferences, the
pre-operation, post-operation, and inter-operation, in or-
der to make solutions to BGDM. A numerical example is
given in Section 4 to illustrate the procedures of three
extended models and their differences. At last, Section 5
concludes this paper.
2. Review of Related Foundations
2.1. Basics of bba
In a specific application domain, the DST first defines Ω,
called the frame of discernment, containing N exhaustive
and exclusive hypotheses. Let 2Ω denote the power set
composed of 2N propositions of A such that A⊆Ω.
Definition 1. Let Ω denote a frame of discernment, and
S be a piece of arbitrary evidence source (ES) on Ω. Thus,
the bba of ES is defined by m: 2Ω→ [0, 1]. This function

12 Chao Fu
verifies the following properties [5, 23]:
()
AmA ⊆Ω
∑=1. (1)
In Shafer’s original definition, m is called basic prob-
ability assignment (bpa) [23] with condition m (Ø) =0.
However, since transferrable belief model (TBM) was
proposed as a model of uncertainty [28], condition m (Ø)
=0 has been omitted. Subsets A of Ω such that m (A)>0 are
called focal elements of m.
Definition 2. Let a power set on Ω be defined as 2Ω=
(B1, B2, …, Br), where r=|2Ω|, the cardinality of 2Ω. Sup-
pose bbai (1≤i≤n) represents the distribution on 2Ω, thus
bbai = (xi1,xi2,…,xir) satisfies:
xij≥0, 0≤j≤r-1, (2)
1
0
r
ij
j
x
−
=
∑=1, i=1, 2, …, n. (3)
Given A⊆Ω, the mass m(A) represents the belief that
supports A, and that, due to lack of the information and
knowledge, does not support any strict subset of A.
Let m1 and m2 be two bbas defined on Ω. Satisfying the
closed world assumption, the normalized Dempster’s rule
of combination is defined as [5,23]
),()())(( 21,,21 CmBmkAmm ACBCB
∑=Ω⊆
∗=⊗ I (4)
where 1
12
,,
1()()
BCB C
K
mBmC
−
⊆Ω =∅
=−
∑I, (5)
(m1⊗m2)(Ø)=0. (6)
Here, ,,1 2
() ()
BCB CmBmC ⊆Ω =∅
∑I is the mass of the
combined belief allocated to the empty-set before nor-
malization. Dempster’s rule is meaningful and can be
applied only when ,,1 2
() ()
BCB CmBmC ⊆Ω =∅
∑I≠1.
2.2. Basics of TOPSIS
2.2.1. MAD M.
MADM problems are a class of decision problems simply
denoted by
12
11112 1
22122 2
12
n
n
n
mm mmn
CC C
A
vv v
A
vv v
A
vv v
L
L
L
MMMMM
L
, (7)
where Ai (1≤i≤m) denotes the ith alternative, Cj (1≤j≤n)
denotes the jth attribute, and vij (1≤i≤m, 1≤j≤n) de-
notes the assessment of DM to the attribute Cj of alterna-
tive Ai.
Suppose W=(w1, w2, …, wn) such that 1
n
j
jw =
∑=1 is a
weight vector, where wj denotes the weight of Cj.
MADM problem solving includes:
(a) Construct the attribute set of system assessment and
correlate system performance and objective;
(b) Confirm the available alternative set for imple-
menting the objective;
(c) Evaluate all alternatives according to the attribute
set and give vij (1
≤ i ≤ m, 1 ≤ j ≤ n).
(d) Apply normalized analysis methodologies to
MADM problems;
(e) Make choice of the best alternative;
(f) Collect new information and start with a new deci-
sion procedure for MADM problems if the resulting al-
ternative can not be accepted.
Steps (a) and (e) orient to DM, but others to applica-
tions. In Step (d), DM expresses his/her preference ac-
cording to the relative importance of every attribute, for
example, setting wj.
2.2.2. TOPSIS
The TOPSIS is an important practical technique to solve
MADA problems originating from the concept of a dis-
placed ideal point from which the compromise solution
has the shortest distance [36]. In the view of Hwang and
Yoon [15], the rating of alternative depends on the short-
est distance from the positive ideal solution (PIS) and the
farthest distance from the negative ideal solution (NIS) or
nadir. Compared with the Analytic Hierarchy Process
(AHP) [22], the TOPSIS fits the cases with a large num-
ber of attributes and alternatives.
In [15], Hwang and Yoon partition attributes into three
classes: benefit ones, cost ones and non-monotonic ones.
The different classes of attributes correspond to different
normalization methods in order to fit different real-world
situations, i.e. the vector normalization, the linear nor-
malization, and the non-monotonic normalization.
Practically, the TOPSIS and its extensions are used to
solve many theoretical and real-world problems, such as
decision making with fuzzy data [16] or interval data
[17], decision support analysis for material selection of
metallic bipolar plates [24], evaluating initial training
aircraft under a fuzzy environment [29], or in-
ter-company comparison [6].
A general flow of TOPSIS involves:
1) Normalize decision matrix V= (vij)m×n.
The decision matrix V is transformed to a normalized
matrix R by
∑−
=m
kkj
ij
ij
v
v
r
1
2
(1≤i≤m, 1≤j≤n), where
rij is the normalized one of vij.
2) Calculate weighted decision matrix Z=(zij)m×n.

Extended TOPSISs for Belief Group Decision Making 13
The normalized matrix R is transformed to a weighted
decision matrix Z such that zij=wj·rij (1≤i≤m, 1≤j≤n),
where wj denotes the weight of Cj such that 1
n
j
jw =
∑=1.
3) Determine PIS and NIS.
The PIS and NIS are respectively
A+= {1
z+,2 z+, …, n z+}={( max
j
zij| j∈Ωb), (min
j
zij|
j∈Ωc)},
A
-
= {1 z−,2 z−, …, n z−}={( min
j
zij| j∈Ωb), (max
j
zij|
j∈Ωc)},
where Ωb and Ωc are benefit attribute set and cost attribute
set, respectively.
4) Compute the separation measures of each alternative
from the PIS and NIS.
The separation measures of each alternative from the
PIS and NIS are respectively
2
1()
n
iijj
j
Dzz
++
=
=−
∑, i=1, 2, …, m,
2
1()
n
iijj
j
Dzz
−−
=
=−
∑, i=1, 2, …, m.
5) Calculate the closeness coefficient of each alterna-
tive.
The closeness of each alternative can be defined as
RCi=i
ii
D
D
D
−
+−
+
，i=1, 2, …, m.
6) Rank the preference order.
The alternative set denoted by Ai (1≤i≤m) is ranked by
means of RCi, which indicates what the best alternative is.
2.3. Discussion
The original TOPSIS has the ability to effectively solve
general MADM problems for one DM, which can easily
extended to deal with the situation of GDM.
In the work of Shih et al. [25], they constructed an in-
ternal extended model of TOPSIS for GDM, in which the
steps were updated involving the decision matrix nor-
malization, distance measures, and aggregation operators.
One can obviously realize that the internal model never
fits external extensions of TOPSIS associated with the
pre-operation and post-operation. Furthermore, it is not
suitable for the internal extension of TOPSIS in this
study, where uncertain group evaluations are represented
by bbas.
In Section 3, three extended models for BGDM, re-
cently researched by Fu etc. in [10-12], are elaborated step
by step, corresponding to the pre-operation,
post-operation, and inter-operation.
3. Solutions to Belief Group Decision Making
According to the classes of group preference aggregation
proposed by Shih et al. [25], we extend the original
TOPSIS to be available for BGDM situation by three
approaches, corresponding to the pre-operation,
post-operation, and inter-operation. Three extended
TOPSIS models are respectively named as pre-model,
post-model, and inter-model. The detailed procedures of
the three models are interpreted as follows.
3.1. Pre-model
The pre-model is composed of the following steps.
Step 1: Construct initial group belief decision matrices
(BDMs).
The initial BDM of each DM can be defined as fol-
lows:
12
11112 1
22122 2
12
n
tt t
n
tt t
n
tt t
mm mmn
CC C
A
yyy
A
yy y
A
yyy
L
L
L
MM MMM
L
(8)
where Ai (1≤i≤m) denotes the ith alternative, Cj (1≤j≤n)
denotes the jth attribute, and t
ij
y
(1≤i≤m, 1≤j≤n,
1≤t≤T) denotes the belief assessment of DM t to the
attribute Cj of alternative Ai. Let Ωj (1≤j≤n) be the frame
of discernment used to generate the assessments on the
attribute Cj. In terms of Definition 2, we have t
ij
y
=
j
t
i
B
Ω
=12
(, ,,)
j
tt t
ii ir
bb bK, where|2 |
j
j
rΩ =.
Convenient to decide the PISB and NISB, the distribu-
tion of power set on Ωj is specified in Definition 3.
Definition3. Let Ωj be the frame of discernment used
to generate the assessments on the attribute Cj (1≤j≤n),
and 12
2(,,,)
j
j
r
B
BB
Ω=K be the distribution of an arbitrary
power set on Ωj, where |2 |
j
j
rΩ =. Suppose the cardi-
nality of Bk is increasing along the increase of k. Fur-
thermore, we assume B1 = Ø (empty-set), B2 and B3 re-
spectively correspond to the single positive ideal element
(SPIE) and the single negative ideal element (SNIE) of
Ωj.
The original TOPSIS requires a uniform dimension for
the assessments on every quantitative attribute. The three
extensions of TOPSIS for BGDM situation are also con-
strained by this requirement. That is to say, the various
frames, Ωj (1≤j≤n), have to be transformed to a unified
frame ΩC so that every attribute can be assessed in a uni-
form, consistent and compatible manner.
The transformation from Ωj (1≤j≤n) to ΩC is stipulated

14 Chao Fu
as Proposition 1.
Proposition1. Let Ωj be the frame of discernment used
to generate the assessments on the attribute Cj (1≤j≤n).
The assessments on Ωj can be equivalently and rationally
transformed to the ones on a common frame of discern-
ment ΩC.
In fact, Proposition 1 is clearly correct since two tech-
niques, a rule based one and a utility based one, are in-
vestigated to accomplish the transformation in Proposition
1 [35].
From Proposition 1, t
ij
y in Eq (8) can be transformed
to a distribution on ΩC. Therefore, the belief attribute
evaluations of each DM to each alternative are unified in
the set of distributions on ΩC. In the following, we sup-
pose t
ij
y denotes a distribution on ΩC.
Step 2: Aggregate group BDMs to form a total BDM.
From Step 1, we know the BDM of each DM as de-
fined in Eq (8). With the normalized Dempster’s rule of
combination [5, 23], group BDMs are combined to form a
total BDM. Let the total BDM be defined in the follow-
ing:
12
11112 1
221222
12
n
n
n
mm mmn
CC C
Ax xx
Ax xx
Ax xx
L
L
L
MMMMM
L
(9)
where xij=j
i
B
Ω=12
(, ,,)
C
ii ir
bb bK, |2 |
C
C
rΩ =, 1 ≤ i ≤ m,
1≤j≤n. Given any element xij in the total BDM, we
have 1
Tt
ij ij
t
xy
=
=⊗ , where the operator ⊗ denotes the
normalized Dempster’s rule of combination as specified
in Eqs (4) to (6). Here, we suppose all experts have the
same importance.
Step 3: Normalize the total BDM.
Different from the original TOPSIS, xij is not a real
number but a normalized distribution on ΩC, the Step can
be omitted.
Step 4: Assign a total weight vector W to the attribute
set.
Let t
W denote the weight vector of each DM as-
signed to the attribute set. We have
t
W=12
(, ,,)
tt t
n
ww wK, 1≤t≤T, 1
nt
j
jw =
∑=1. The total
weight vector W can be defined as the arithmetical mean
of all t
W (1≤t≤T), which is W= (w1, w2, …, wn) such
that
1
1Tt
j
j
t
ww
T=
=∑, 1≤j≤n. (10)
Step 5: Determine the total PISB and NISB.
Before determining the total PISB and NISB, first of
all we define the PISB and NISB in Definition 4, owing
to the distribution specification in Definition 3.
Definition4. Based on the specification in Definition 3,
given the attribute Cj (1≤j≤n), no matter whether it is
the benefit attribute or the cost attribute, its PISB and
NISB are respectively
C
C
r2
1r
(0,1,0, ,0)
−
×
678
K and
C
C
r3
1r
(0,0,1, 0,,0 )
−
×
678
K.
According to Definition 4, by combining the PISB and
NISB of each attribute, we achieve the total PISB and
NISB of total BDM.
Step 6: Calculate the separation measures of each al-
ternative from the total PISB and NISB.
From Step 5, the total PISB and NISB can be respec-
tively denoted by
()
+
×C rn
S1=(0,1,0, ,0,
C
r
6
4748
K ,L 0,1,0, ,0)
C
r
647 48
K
and
()
−
×C rn
S1=(0,0,1,0, ,0,
C
r
6
447 448
K ,L 0,0,1, 0,, 0)
C
r
6447 448
K.
Furthermore, in order to precisely reflect the preference
of each DM and the physical implication of each subset of
the distribution on 2C
Ω
when calculating the separation
measures of each alternative from the PISB and NISB, we
define the positive preference vector (PPV)
(1,, ,,
C
ttt
kr
β
ββ
+
++
KK ) and the negative preference vector
(NPV) (1,, ,,
C
ttt
kr
βββ
−
−−
KK
) of each DM for the distribu-
tion on C
Ω
2 where
1
1
C
r
t
k
k
β
+
=
=
∑,
1
1
C
r
t
k
k
β
−
=
=
∑, |2 |
C
C
rΩ =.
Through ordered comparison of any two different subsets
of the distribution onC
Ω
2 the PPV and NPV of DM can
be achieved. We postulate 0
t
k
β
+>,0 t
k
β
−>, if k>1, and
0
tt
kk
ββ
+−
=
=, if k=1, so as to keep all available infor-
mation. Let the positive group preference vector (PGPV)
and negative group preference vector (NGPV) respec-
tively be (1,, ,,
C
kr
β
ββ
+
++
KK) and (1,,,,
C
kr
β
ββ
−−−
KK) such
that
1
1
C
r
k
k
β
+
=
=
∑,
1
1
C
r
k
k
β
−
=
=
∑, we thus have
1
1T
t
kk
t
T
β
β
+
+
=
=
∑
, (11)

Extended TOPSISs for Belief Group Decision Making 15
1
1T
t
kk
t
T
β
β
−−
=
=∑. (12)
The PPV and NPV can effectively avoid the possible
paradoxes between calculating results and the fact of
DM’s preference as well as physical implications of
worlds in ΩC.
Hence, the separation measures of each alternative
from the total PISB and NISB are expressed as
2
)1((1
11
)( +
+−
=
+
=
+−= ∑∑ krjik
r
k
k
n
j
ji C
C
SbwD β (13)
and
2
)1((1
11
)( −
+−
=
−
=
−−= ∑∑ krjik
r
k
k
n
j
ji C
C
SbwD β (14)
where 1≤i≤m, |2 |
C
C
rΩ =, with the approach of
Euclidian distance [9].
Step 7: Compute the closeness coefficient *
i
E of
each alternative for group.
The closeness coefficient of each alternative can be
defined as
*
i
E
=/( )
ii i
DD D
−− +
+(1≤i≤m). (15)
The larger the value of*
i
E
, the better the alternative.
Step 8: Rank the preference order.
In terms of*
i
E
, a set of alternatives will be ranked in
an incremental order representing group preferences.
3.2. Post-model
The post-model is partially the same as the pre-model.
After the procedure of original TOPSIS, the rank of each
alternative representing group preferences is determined,
aided by one of social choice functions [14], such as the
Borda function in this paper.
Step 1: Construct initial group BDMs.
The Step is the same as Step 1 of pre-model.
Step 2: Normalize the BDM of each DM.
Same as Step 3 of pre-model, the Step can be omitted.
Step 3: Assign the weight vector Wt to the attribute set
for each DM.
We suppose Wt denotes the weight vector of DM t as-
signed to the attribute set, where Wt=12
(, ,,)
tt t
n
ww wK,
1≤t≤T, 1
nt
j
jw
=
∑=1.
Step 4: Determine the PISB and NISB of each DM.
As specified in Definition 3, the PISB and NISB of each
DM are respectively denoted by
()
+
×
t
rn C
S1= (0,1,0, ,0,
C
r
6
4748
K ,L 0,1, 0,, 0)
C
r
64748
K
and
()
−
×
t
rn C
S1= (0,0,1,0, ,0,
C
r
6
447448
K ,L 0, 0,1, 0,, 0)
C
r
6447448
K, where
1≤t≤T.
Step 5: Calculate the separation measures of each al-
ternative from the PISB and NISB of each DM.
Similar to Step 6 of pre-model, the separation measures
of each alternative from the PISB and NISB for each DM
are expressed as
2
)1((1
11
)( +
+−
=
+
=
+−=∑∑ t
krjik
r
k
t
k
n
j
t
j
t
iC
C
SbwD β (16)
and
2
)1((1
11
)( −
+−
=
−
=
−−=∑∑ t
krjik
r
k
t
k
n
j
t
j
t
iC
C
SbwD β (17)
where 12
(,, ,)
C
ii ir
bb bK=t
ij
y, 1≤i≤m, 1≤t≤T,
|2 |
C
C
rΩ =.
Step 6: Compute the closeness coefficient *t
i
E of each
alternative for each DM.
The closeness coefficient of each alternative for each
DM can be defined as
*t
i
E
=/( )
ttt
ii i
DDD
−
−+
+, (18)
where 1≤i≤m, 1≤t≤T.
Step 7: Rank the preference order of each DM.
In terms of*t
i
E
, a set of alternatives will be ranked in
an incremental order representing the preference of each
DM, where 1≤t≤T.
Step 8: Give the Borda score of each alternative ac-
cording to the preference order of each DM.
Suppose the preference order of DM t is
1
ttt
im
BBBfKf fKf , where t
i
B (1≤i≤m) is the same
as t
j
A
(1≤j≤m). The Borda score of 1
t
B is m
- 1, the
ones of 2
t
B and t
m
B are respectively m
- 1 and 0, and
the rest may be deduced by analogy.
Step 9: Aggregate the Borda score of each alternative
given by each DM.

16 Chao Fu
Let the Borda score vectors of each alternative repre-
senting the preference of DM t and group preferences be
respectively (1,,,,
ttt
im
SSSKK) and (1,,,,
im
SSSKK).
We have
1
T
t
ii
t
SS
=
=∑, 1≤i≤m. (19)
Step 10: Rank the preference order for group.
According to (1,,,,
im
SSSKK), we rank the prefer-
ence order of a set of alternatives for group.
3.3. Inter-model
The inter-model is similar to the internal TOPSIS model
of Shih et al. [25]. It combines the individual separation
measures of each alternative from the PISB and NISB to
form group measures within the TOPSIS procedure.
The first five Steps of inter-model are the same as Steps
1 to 5 of post-model.
Step 6: Combine the individual measures of each al-
ternative from the PISB and NISB to form group meas-
ures.
From Step 5 of post-model, we achieve the individual
measures of each alternative from the PISB and NISB,
which are respectively t
i
D
+ and t
i
D
− (1≤i≤m, 1≤t≤T).
Thus, the group measures of each alternative are respec-
tively
1
Tt
ii
t
D
D
++
=
=⊕ (20)
and
1
Tt
ii
t
D
D
−−
=
=⊕ . (21)
The operator ⊕ can be the arithmetical mean, the
geometric mean, or their modifications. In this paper, the
arithmetical mean is our choice.
Steps 7 and 8 are the same as Steps 7 and 8 of
pre-model.
As mentioned above, three extended models are similar
to each other in many Steps. The main differences lie in
the aggregation of group preferences.
In the pre-model, thanks to two strategies of Dempster’s
rule modification (e.g. [8, 19, 26-27, 32-33]) and source
modification (e.g. [7, 13, 21]) aiming at combining con-
flicting beliefs, the preference conflicts between different
DMs can be effectively dealt with. In the post-model,
some social choice functions [14] can be selected to
guarantee group preferences aggregation is rational and
available in different applications. In the inter-model, the
arithmetical mean, the geometric mean, or their modifi-
cations are used to aggregate the individual separation
measures of each alternative from the PISB and NISB.
In practice, how to select the appropriate extended
model depends on how to select the appropriate approach
to aggregating group preferences, which is the most
suitable one for real-world problems.
4. Numerical Example
To clearly illustrate the procedures of three extended
models, a numerical example is shown as follows.
From Tables 1 to 3, one can know initial group BDMs,
and the preference vectors and weight vector of each DM.
There are two attributes, three alternatives, and three
DMs in this example. Two attributes C1 and C2 are the
benefit one and the cost one, respectively. Suppose Ω1=
{good, common}, Ω2= {small, big, common}, ΩC= {first,
second, third}, according to Proposition 1, the assess-
ments on Ω1 and Ω2 can be equivalently transformed to the
ones on ΩC. In terms of Definition 3, the power set on ΩC
is {{Ø}, {first}, {third}, {second}, {first, third}, {first,
second}, {second, third}, {first, second, third}}.
As specified in Definition 4, the PISB and NISB are
respectively (0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0) and (0,0,1,0,
0,0,0,0,0,0,1,0,0,0,0,0). The decision procedures of three
extended models will be presented as follows.
In the pre-model, group belief evaluations are firstly
combined to form the total BDM displayed in Table 4,
with the normalized Dempster’s rule of combination.
Afterwards, according to Eq (10), the total weight
vector W= (0.6, 0.4) is generated from the weight vectors
in Table 3. Based on the data in Table 2, the PGPV and
NGPV are computed respectively as (0,0.03,0.207,0.092,
0.207,0.05,0.207,0.207) and (0,0.384,0.055,0.163,0.055,
0.233,0.055,0.055), in terms of Eqs (11) and (12).
With the above results, the total separation measures
and the closeness coefficient of each alternative are ob-
tained in Table 5, according to Eqs (13) to (15).
From Table 5, the preference order of three alternatives
is known to be A1fA3fA2, where the notation “f”
means “prior”.
In the post-model, first of all the individual separation
measures and the closeness coefficient of each alternative
are computed in Table 6.
The Borda score and rank of each alternative for group
are generated from the data in Table 6 and shown in Table
7.
According to Table 7, three alternatives are ranked by
the preference order A1fA2=A3.
In the inter-model, the separation measures and close-
ness coefficient of each alternative for group are achieved
in Table 8, on the basis of the data in Table 6.
Three alternatives are ranked with the preference order
A1fA2fA3 according to Table 8.

Extended TOPSISs for Belief Group Decision Making 17
The three preference orders corresponding to three ex-
tended models are pair-wise different. The mediator and
the requirements of a real application decide which order
is the best one and which extended model should be ap-
plied. Especially, if the mediator only wants to know the
best alternative, it is unnecessary to differentiate the three
orders.
Table 1. Initial group BDMs
C1 C2
DM1 (0,0.6,0,0,0,0.4,0,0) (0,0.3,0.2,0,0,0.5,0,0)
DM2 (0,0.5,0,0.2,0,0.3,0,0) (0,0.5,0.2,0,0,0,0.3,0) A1
DM3 (0,0.4,0,0.2,0,0.4,0,0) (0,0.4,0,0.4,0,0.2,0,0)
DM1 (0,0.2,0,0.5,0,0,0.3,0) (0,0.6,0.2,0,0,0.2,0,0)
DM2 (0,0.3,0,0.5,0,0.2,0,0) (0,0.4,0.1,0,0,0,0.5,0) A2
DM3 (0,0.4,0,0.3,0,0.3,0,0) (0,0.5,0.3,0,0,0.2,0,0)
DM1 (0,0.2,0,0.8,0,0,0,0) (0,0.2,0.4,0,0,0,0.4,0)
DM2 (0,0.7,0,0,0,0.3,0,0) (0,0.4,0.2,0.4,0,0,0,0) A3
DM3 (0,0.6,0,0.1,0,0.3,0,0) (0,0.2,0.6,0,0,0.2,0,0)
Table 2. The preference vectors of each DM
18
(,,)
β
β
++
K 18 (,,)
β
β
−
−
K
DM1 (0,0.04,0.2,0.1,0.2,0.06,0.2,0.2) (0,0.4,0.05,0.15,0.05,0.25,0.05,0.05)
DM2 (0,0.03,0.2,0.12,0.2,0.05,0.2,0.2) (0,0.3,0.09,0.14,0.09,0.2,0.09,0.09)
DM3 (0,0.02,0.22,0.06,0.22,0.04,0.22,0.22) (0,0.45,0.025,0.2,0.025,0.25,0.025,0.025)
Table 3. The w e ight vector of each DM
w1 w2
DM1 0.5 0.5
DM2 0.7 0.3
DM3 0.6 0.4
Table 4. The total group BDM
C1 C2
A1 (0,0.83,0,0.1,0,0.07,0,0) (0,0.73,0,0.27,0,0,0,0)
A2 (0,0.17,0,0.83,0,0,0,0) (0,0.8,0.13,0.07,0,0,0,0)
A3 (0,0.65,0,0.35,0,0,0,0) (0,0.2,0,0.8,0,0,0,0)
Table 5. The separation measures and closeness coefficient
of each alternative in the pre-model
D+ D- E*= D-/( D-+ D+) rank
A1 0.06911 0.54954 0.8883 1
A2 0.2291 0.4715 0.673 3
A3 0.2005 0.46065 0.6967 2

18 Chao Fu
Table 6. The separation measures and the closeness coefficient
of each alternative in the post-model
S+ S- E*
DM1 0.17117 0.42691 0.7138
DM2 0.14768 0.41741 0.7387 A1
DM3 0.13023 0.37762 0.7436
DM1 0.19925 0.38341 0.658
DM2 0.227 0.39373 0.6343 A2
DM3 0.14241 0.36932 0.7217
DM1 0.29933 0.31937 0.5162
DM2 0.12822 0.46573 0.7841 A3
DM3 0.20465 0.37376 0.6462
Table 7. The Borda score and rank of each alternative
Borda score rank
A1 5 1
A2 2 2
A3 2 2
Table 8. The separation measures and closeness coefficient
of each alternative in the inter-model
D+ D- E*= D-/( D-+ D+) rank
A1 0.14969 0.407310.7313 1
A2 0.18955 0.382150.6684 2
A3 0.21073 0.386290.647 3
5. Conclusions
Through representing the uncertain attribute evaluations of
a group of DMs to alternatives by bbas, the common GDM
is extended to the BGDM. To solve the MADA problems
in the situation of BGDM, we develop three extended
TOPSIS models, the pre-model, post-model, and in-
ter-model, associated with three approaches to aggregating
group preferences, the pre-operation, post-operation, and
inter-operation.
For the BGDM, three extended models are elaborated
step by step, based on the equivalent transformation of the
assessments on different frames of discernment, the PISB
and NISB, and the PPV and NPV of each DM. Further-
more, a numerical example clearly illustrates the proce-
dures of three extended models.
The reliability of experts may be an important factor to
influence our method. If a group of experts have different
reliability, their bbas may be discounted [23] before used
in the three models. The discounting approach is intro-
duced in the original work of Shafer [23]. In practical
applications, how to decide the reliability of experts may
be a problem difficult to solve [19].
The computational complexity may be a problem for our
method is on the power set of a frame of discernment. In
fact, the numerical examples in Section 4 are solved by the
program made by Microsoft Visual C++ 6.0 within several
seconds. By testing randomly selected data, we find that
when |Ω|<13, the solutions can be obtained within several
seconds. Note that for the MADA problems in the situation
of BGDM, |Ω|<13 is generally enough to provide the
satisfactory service for experts. If |Ω| is too large, experts
will have difficulties to make decisions. Therefore, the
computational complexity of our method can be effec-
tively solved by the computer program and the real con-
straints of experts’ decision making.
6. Acknowledgement
This research is supported by the National Natural Science
Foundation of China (No. 70631003 and 90718037) and

Extended TOPSISs for Belief Group Decision Making 19
the Foundation of Hefei University of Technology (No.
081104F).
We would like to thank the anonymous reviewers for
their constructive comments helping us to improve this
paper considerably.
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AUTHOR’S BI OG R A PHY
Chao Fu received his M.S. degree in Mechanical and Electronic Engineering from Huazhong University of Science and
Technology, Wuhan, Hubei, China in 2003. He joined Hefei University of Technology, where he is currently a lecturer at
School of Management. He is a researcher of Key Laboratory of Process Optimization and Intelligent Decision-making,
Ministry of Education, in the same school. His research interests include decision science and technology, information
systems and engineering, and the simulation of complex decision task.