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A method for multiple attribute decision making with incomplete weight
information under uncertain linguistic environment
Yejun Xu
a,b,*
, Qingli Da
a
a
College of Economics and Management, SouthEast University, Nanjing 210096, China
b
College of Economics and Management, Nanjing Forestry University, Nanjing 210037, China
article info
Article history:
Received 23 September 2007
Accepted 28 March 2008
Available online 4 April 2008
Keywords:
Multiple attribute decision making under
linguistic setting
Uncertain linguistic variable
Deviation degree
Possibility degree
Ideal point
abstract
The multi-attribute decision making problems are studied, in which the information about the attribute
values take the form of uncertain linguistic variables. The concept of deviation degree between uncertain
linguistic variables is defined, and ideal point of uncertain linguistic decision making matrix is also
defined. A formula of possibility degree for the comparison between uncertain linguistic variables is pro-
posed. Based on the deviation degree and ideal point of uncertain linguistic variables, an optimization
model is established, by solving the model, a simple and exact formula is derived to determine the attri-
bute weights where the information about the attribute weights is completely unknown. For the infor-
mation about the attribute weights is partly known, another optimization model is established to
determine the weights, and then to aggregate the given uncertain linguistic decision information, respec-
tively. A method based on possibility degree is given to rank the alternatives. Finally, an illustrative
example is also given.
Ó2008 Elsevier B.V. All rights reserved.
1. Introduction
Multiple attribute decision making is a prominent area of re-
search in normative decision theory. This topic has been widely
studied [1,4,6,12,18,24]. It generally involves the following three
phases:
(1) It needs collecting the information about attribute weights
and attribute values.
(2) It involves weighted aggregation of the attribute values
across all attributes for each alternative to obtain an overall
value.
(3) It orders the overall values to obtain the best alternative(s).
In the real world, the decision maker (DM) may have vague
knowledge about the preferences degree of one alternative over
another. Furthermore, it is too complex or too ill-defined to be
amenable for description in conventional quantitative expressions.
It is more suitable to provide their preferences by means of linguis-
tic variables rather than numerical ones [8,9,13,19,21,25]. Many
approaches have been proposed for aggregating linguistic informa-
tion up to now [3,5,11]. An ordinal structure of the linguistic term
sets, has been developed in Ref. [5]. An approximate computations
over the linguistic variables has been developed in Ref. [3]. Herrera
and Martínez [11] have developed a fuzzy linguistic representation
model, which represents the linguistic information with a pair of
values called 2-tuple, composed by a linguistic term and a number.
Xu [20] has developed a direct approach to decision making with
linguistic preference relations. In some situations, however, the
attribute values take the form of uncertain linguistic variables
rather than numerical one, the information about attribute weight
is completely unknown or partly known. The above approaches
will fail in dealing with the situations in which the decision infor-
mation takes the form of uncertain linguistic variables. Therefore,
it is necessary to pay attention to this issue.
Technique for order performance by similarity to ideal solution
(TOPSIS), one of the known classical MCDM method, was first
developed by Hwang and Yoon [12] for solving a MCDM problem.
It based upon the concept that the chosen alternative should have
the shortest distance from the ideal solution (IS). In this paper, we
extend the TOPSIS method to uncertain linguistic environment
[22,23], to overcome the above limitation, and we finally get the
attribute weights. In order to do this, this paper is structured as fol-
lows. Section 2gives the concept of the uncertain linguistic vari-
ables and some operational laws of uncertain linguistic variables.
In order to compare uncertain linguistic variables, we present a
formula of possibility degree of uncertain linguistic variables and
propose the properties of the possibility degree. In Section 3we de-
fine some useful concepts. In this section, we extend the concepts
of TOPSIS to the uncertain linguistic environment and define the
concept of the distance of two uncertain linguistic variables. In
0950-7051/$ - see front matter Ó2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.knosys.2008.03.034
* Corresponding author. Address:College of Economics and Management, South-
East University, Nanjing 210096, China. Tel.: +86 25 85427377; fax: +86 25
85427972.
E-mail address: xuyejohn@163.com (Y. Xu).
Knowledge-Based Systems 21 (2008) 837–841
Contents lists available at ScienceDirect
Knowledge-Based Systems
journal homepage: www.elsevier.com/locate/knosys
Section 4, based on the deviation degree and ideal point of uncertain
linguistic variables, an optimization model is established, by solving
the model, a simple and exact formula is derived to determine the
attribute weights where the information about the attribute weights
is completely unknown. For the information about the attribute
weights is partly known, another optimization model is established
to determine the weights, then we utilize the uncertain linguistic
weighted average (ULWA) operator to aggregate the uncertain
linguistic variables corresponding to each alternative, respectively.
A method based on the possibility degree of uncertain linguistic
variables to rank the alternatives. Section 5, a practical application
of the developed method to evaluate the technological innovation
capability of enterprises has also been given to show the effective-
ness of the proposed method. Section 6, we conclude the paper.
2. Preliminaries
The linguistic approach is an approximate technique which rep-
resents qualitative aspects as linguistic values by means of linguis-
tic variables [10,11,20,22,25]. Suppose that S={s
i
|i=t,...,t}isa
finite and totally ordered discrete term set, where s
i
represents a
possible value for a linguistic variable. For example, a set of nine
terms Scould be
S¼fs
4
¼extremely poor;s
3
¼very poor;s
2
¼poor;
s
1
¼slightly poor;s
0
¼fair;s
1
¼slightly good;s
2
¼good;
s
3
¼very good;s
4
¼extremely goodg
Obviously, the mid linguistic label s
0
represents an assessment of
‘‘indifference”, and with the rest of the linguistic labels beings
placed symmetrically around it.
In these cases, it is usually required that there exist the follow-
ing [8,20,22,23]:
(1) The set is ordered: s
i
Ps
j
if iPj;
(2) There is the negation operator: neg(s
i
)=s
j
such that i+j=0;
(3) Max operator: max(s
i
,s
j
)=s
i
if s
i
Ps
j
;
(4) Min operator: min(s
i
,s
j
)=s
i
if s
i
6s
j
.
To preserve all the given information, we extend the discrete
term set Sto a continuous term set [21,22] S¼fs
a
ja2½t;tg.If
s
a
2S, then we call s
a
an original linguistic term, otherwise, we call
s
a
a virtual linguistic term. In general, the decision maker used the
original linguistic terms to evaluate alternatives, and the virtual
linguistic terms can only appear in operation.
Definition 1. [23] Let ~
s¼½s
a
;s
b
, where s
a
;s
b
2S;s
a
and s
b
are the
lower and the upper limits, respectively, we then call ~
sthe
uncertain linguistic variable.
Let e
Sbe the set of all uncertain linguistic variables. Consider any
three uncertain linguistic variables ~
s¼½s
a
;s
b
;~
s
1
¼½s
a
1
;s
b
1
, and
~
s
2
¼½s
a
2
;s
b
2
, then their operational laws are defined as:
(1) ~
s
1
~
s
2
¼½s
a
1
;s
b
1
½s
a
2
;s
b
2
¼½s
a
1
s
a
2
;s
b
1
s
b
2
¼
½s
a
1
þa
2
;s
b
1
þb
2
;
(2) k~
s¼k½s
a
;s
b
¼½ks
a
;ks
b
¼½s
ka
;s
kb
, where k2[0,1];
(3) ~
s
1
~
s
2
¼~
s
2
~
s
1
;
(4) kð~
s
1
~
s
2
Þ¼k~
s
1
k~
s
2
, where k2[0,1];
(5) ðk
1
þk
2
Þ~
s¼k
1
~
sk
2
~
s, where k
1
,k
2
2[0,1].
In order to compare uncertain linguistic variables, we give the
following definition:
Definition 2. Let ~
s
1
¼½s
a
1
;s
b
1
and ~
s
2
¼½s
a
2
;s
b
2
2e
S, be two
uncertain linguistic variables, and let lenð~
s
1
Þ¼b
1
a
1
;
lenð~
s
2
Þ¼b
2
a
2
;winð~
s
1
Þ¼b
1
þa
1
;winð~
s
2
Þ¼b
2
þa
2
, then the
degree of possibility of ~
s
1
P~
s
2
is defined as
pð~
s
1
P~
s
2
Þ¼min max 1
2
winð~
s
1
Þwinð~
s
2
Þ
lenð~
s
1
Þþlenð~
s
2
Þþ1
;0
;1
ð1Þ
From Definition 2, we can easily get the following properties of the
degree of possibility.
Theorem 1. Let ~
s
1
¼½s
a
1
;s
b
1
;~
s
2
¼½s
a
2
;s
b
2
and ~
s
3
¼½s
a
3
;s
b
3
2e
Sbe
three uncertain linguistic variables, then
(1) 06pð~
s
1
P~
s
2
Þ61;
(2) pð~
s
1
P~
s
2
Þ¼1, if and only if b
2
6a
1
;
(3) pð~
s
1
P~
s
2
Þ¼0, if and only if b
1
6a
2
;
(4) pð~
s
1
P~
s
2
Þþpð~
s
2
P~
s
1
Þ¼1. Especially, pð~
s
1
P~
s
1
Þ¼1=2;
(5) pð~
s
1
P~
s
2
ÞP1=2if and only if a
1
+b
1
Pa
2
+b
2
. Especially,
pð~
s
1
P~
s
2
Þ¼1=2, if and only if a
1
+b
1
=a
2
+b
2
;
(6) pð~
s
1
P~
s
2
ÞP1=2and pð~
s
2
P~
s
3
ÞP1=2, then pð~
s
1
P~
s
3
Þ
P1=2.
3. Uncertain linguistic weighted averaging operator
For an uncertain multiple attribute decision making problem, let
X={x
1
,x
2
,...,x
m
} be a discrete set of alternatives, U={u
1
,u
2
,...,u
n
}
be a set of attributes. For each alternative x
i
2X, the decision maker
gives his/her preference value ã
ij
with respect to attribute u
j
2U,
where ã
ij
takes the form of uncertain linguistic variable, that is
~
a
ij
2e
Sð~
a
ij
¼½a
L
ij
;a
U
ij
;a
L
ij
2S,a
U
ij
2SÞ, then all the preference values of
the alternatives consists the decision matrix e
A¼ð
~
a
ij
Þ
mn
.
Definition 3. [23]. Let ULWA: e
S
n
!e
S,if
ULWA
w
ð~
s
1
;~
s
2
;...;~
s
n
Þ¼w
1
~
s
1
w
2
~
s
2
w
n
~
s
n
ð2Þ
where w=(w
1
,w
2
,...,w
n
)
T
is the weighting vector of uncertain lin-
guistic variables ~
s
i
ði¼1;2;...;nÞ, and w
i
2½0;1;i¼1;2;...;
n;P
n
i¼1
w
i
¼1, then ULWA is called the uncertain linguistic
weighted averaging (ULWA) operator. Especially, if w= (1/n,1/
n,...,1/n)
T
, then the ULWA operator is reduced to the ULA operator.
Definition 4. Let e
A¼ð
~
a
ij
Þ
mn
be the uncertain linguistic decision
matrix, ã
i
=(ã
i1
,ã
i2
,...,ã
in
) be the vector of attribute values corre-
sponding to the alternative x
i
,i=1,2,...,m, then we call
~
z
i
ðwÞ¼ULWA
w
ð~
a
i1
;~
a
i2
;...;~
a
in
Þ¼w
1
~
a
i1
w
2
~
a
i2
...w
n
~
a
in
ð3Þ
the overall value of the alternative x
i
, where w=(w
1
,w
2
,...,w
n
)
T
is
the weighting vector of attributes.
The uncertain linguistic multiple attribute decision making
problems generally consist of finding the most desirable
alternative(s) from a given alternative set. If the decision matrix
e
A¼ð
~
a
ij
Þ
mn
and the corresponding weight value of attributes are
known, we can get the overall value of the alternative x
i
by Eq. (3).
TOPSIS (technique for order performance by similarity to ideal
solution) is a useful technique in dealing with multi-attribute or
multi-criteria decision making problems in the real word [12].It
originates from the concept of a displaced ideal point from which
the compromise solution has the shortest distance [2,26]. Hwang
and Yoon [12] further propose that the ranking of alternatives will
be based on the shortest distance from the (positive) ideal solution
(PIS). Therefore, we can know that the TOPSIS technique is a sound
logic that represents the rationale of human choice. In the follow-
ing, we extend the TOPSIS to the uncertain linguistic variables.
Definition 5. Let e
A¼ð
~
a
ij
Þ
mn
be the uncertain linguistic decision
matrix, then the ideal solution of attribute values are defined as
following:
~
a
þ
¼ð
~
a
þ
1
;~
a
þ
2
;...;~
a
þ
n
Þ
838 Y. Xu, Q. Da / Knowledge-Based Systems 21 (2008) 837–841
where ~
a
þ
j
¼½a
þL
j
;a
þU
j
¼½max
i
a
L
ij
;max
i
a
U
ij
;j¼1;2;...;nð4Þ
and a
þL
j
;a
þU
j
are the lower bound and upper bound of ~
a
þ
j
Definition 6. Let ~
s
1
¼½s
a
1
;s
b
1
and ~
s
2
¼½s
a
2
;s
b
2
2e
Sbe two uncer-
tain linguistic variables, then we call
Dð~
s
1
;~
s
2
Þ¼ja
2
a
1
jþjb
2
b
1
jð5Þ
the distance between ~
s
1
and ~
s
2
.
Definition 7. Let e
A¼ð
~
a
ij
Þ
mn
be the uncertain linguistic decision
matrix, ~
a
þ
¼ð
~
a
þ
1
;~
a
þ
2
;...;~
a
þ
n
Þbe the ideal point of attribute values,
defined as Definition 5, then we call
~
zðwÞ¼ULWA
w
ð~
a
þ
1
;~
a
þ
2
;...;~
a
þ
n
Þ¼w
1
~
a
þ
1
w
2
~
a
þ
2
...w
n
~
a
þ
n
ð6Þ
the overall value of ideal point ã
+
.
4. Models and methods
In the real life, there always exist some differences between the
ideal point of attribute values and the vector of attribute values cor-
responding to the alternative x
i
(i=1,2,...,m). By Definitions 3–6,in
what follows we define the distance d
i
between the overall value
~
zðwÞof ideal point and the overall value ~
z
i
ðwÞof the alternative x
i
:
d
i
¼X
n
j¼1
ðDð~
a
þ
j
;~
a
ij
Þw
j
Þ
2
;i¼1;2;...;mð7Þ
Obviously, the smaller d
i
, the better the alternative x
i
will be. Thus, a
reasonable weight vector w
¼ðw
1
;w
2
;...;w
n
Þ
T
should be deter-
mined so as to make all the distances d
i
(i=1,2,...,m) as smaller as
possible, which means to minimize the following distance vector:
dðwÞ¼ðd
1
;d
2
;...;d
m
Þ
In order to do that, we establish the following multiple objective
optimization model:
ðM-1Þmin FðwÞ¼ðd
1
;d
2
;...;d
m
Þ
s:t:X
n
j¼1
w
j
¼1;w
j
P0j¼1;2;...;n
Generally, all the objectives are fairly competitive and there is no
preference relationship among them, therefore the above model
can be transformed into the following goal programming problem:
ðM-2Þmin FðwÞ¼X
m
i¼1
X
n
j¼1
ðDð~
a
þ
j
;~
a
ij
Þw
j
Þ
2
s:t:w
j
P0;X
n
j¼1
w
j
¼1
To solve this model, we construct the Lagrange function:
Fðw;kÞ¼X
m
i¼1
X
n
j¼1
ðDð~
a
þ
j
;~
a
ij
Þw
j
Þ
2
þ2kX
n
j¼1
w
j
1
! ð8Þ
where kis the Lagrange multiplier.
Differentiating Eq. (8) with respect to w
j
(j=1,2,...,n) and k,
and setting these partial derivatives equal to zero, the following
set of equations is obtained:
oF
ow
j
¼2P
m
i¼1
D
2
ð~
a
þ
j
;~
a
ij
Þw
j
þ2k¼0;j¼1;2;...;n
oF
ok
¼P
n
j¼1
w
j
1¼0
8
>
>
>
<
>
>
>
:
ð9Þ
By solving Eq. (9), then we can get:
w
j
¼ k
P
m
i¼1
D
2
ð~
a
þ
j
;~
a
ij
Þ;j¼1;2;...;nð10Þ
X
n
j¼1
w
j
¼1ð11Þ
By solving Eqs. (10) and (12), we get:
k¼ 1
P
n
j¼11
P
m
i¼1
D
2
ð~
a
þ
j
;~
a
ij
Þ
ð12Þ
By solving Eqs. (10) and (11), we can get:
w
j
¼1
P
n
j¼11
P
m
i¼1
D
2
ð~
a
þ
j
;~
a
ij
Þ
X
m
i¼1
D
2
ð~
a
þ
j
;~
a
ij
Þ
.;j¼1;2;...;nð13Þ
which can be used as the weight vector of attributes. Obviously,
w
j
P0, for all j.
In the real world, the information about attribute weights is
incompletely known. Let w=(w
1
,w
2
,...,w
n
)
T
2Hbe the weight
vector of attributes, where w
j
2½0;1;j¼1;2;...;n;P
n
j¼1
w
j
¼1;H
is a set of the known weight information, which can be constructed
by the following forms [2,14–17,21,26], for i–j.
Form 1. A weak ranking: {w
i
Pw
j
};
Form 2. A strict ranking: {w
i
w
j
Pa
i
(>0)};
Form 3. A ranking of differences: {w
i
w
j
Pw
k
w
l
}, for
j–k–l;
Form 4. A ranking with multiples: {w
i
Pa
i
w
j
}, 0 6a
i
61;
Form 5. An interval form: a
i
6w
i
6a
i
+e
i
,06a
i
<a
i
+e
i
61.
Forms 1–2 and Forms 4–5 are well known types of imprecise
information, and Form 3 is ranking of differences of adjacent
parameters obtained by ranking between two parameters, which
can be constructed based on Form 1.
If the information about attribute weights is partly known, and
Eq. (7) is replaced with the following deviation function
d
i
¼X
n
j¼1
Dð~
a
þ
j
;~
a
ij
Þw
j
;i¼1;2;...;mð14Þ
Then, we can establish the following multiple-objective program-
ming model:
ðM-3Þmin FðwÞ¼ð
d
1
;
d
2
;...;
d
m
Þ
s:t:w2H
X
n
j¼1
w
j
¼1;w
j
P0j¼1;2;...;n
Generally, all the objectives are fairly competitive and there is no pref-
erence relationship among them, therefore the above model can be
transformed into the following linear goal programming problem:
ðM-4Þmin X
m
i¼1
k
i
s:t:X
n
j¼1
Dð~
a
þ
j
;~
a
ij
Þw
j
6k
i
;i¼1;2;...;m;
w2H;
X
n
j¼1
w
j
¼1;w
j
P0;j¼1;2;...;n:
From the above analysis, we know that both the models (M-2) and
(M-4) can be used to determine the attribute weights in a multiple
attribute decision making problem with incomplete weight informa-
tion under linguistic environment. Then we can utilize the Eq. (3) to
get the overall value ~
z
i
ðwÞði¼1;2;...;mÞof each alternative. As
~
z
i
ðwÞis still the uncertain linguistic variable, it is difficult to rank them
directly. Therefore, we can utilize the formula (1) to compare each
~
z
i
ðwÞwith all ~
z
i
ðwÞði¼1;2;...;mÞ. For simplicity, we let
p
ij
¼pð~
z
i
ðwÞP~
z
j
ðwÞÞ, then we develop a complementary matrix as
P=(p
ij
)
mm
, where
p
ij
P0;p
ij
þp
ji
¼1;p
ii
¼1
2;i;j¼1;2;...;m
Y. Xu, Q. Da / Knowledge-Based Systems 21 (2008) 837–841 839
Summing all elements in each line of matrix P, we have
p
i
¼X
m
j¼1
p
ij
;i¼1;2;...;mð15Þ
Then we rank ~
z
i
ðwÞði¼1;2;...;mÞin descending order in accor-
dance with the values of p
i
(i=1,2,...,m).
Based on the above model, we develop a practical method for
solving the uncertain linguistic multiple attribute decision making
problems, in which the information about attribute weights is
incompletely known, and the attribute values take the form of
uncertain linguistic variables. The method involves the following
steps:
Step1: Let X={x
1
,x
2
,...,x
m
} be a discrete set of alternatives,
U={u
1
,u
2
,...,u
n
} be a set of attributes, and
w=(w
1
,w
2
,...,w
n
)
T
2Hbe the weight vector of attributes,
where w
j
2½0;1;j¼1;2;...;n;P
n
j¼1
w
j
¼1;His a set of
the known weight information, which can be constructed
by the Forms 1–5. For each alternative x
i
2X, the decision
maker gives his/her preference value ã
ij
with respect to
attribute u
j
2U, where ã
ij
takes the form of uncertain lin-
guistic variable, that is ~
a
ij
2e
Sð~
a
ij
¼½a
L
ij
;a
U
ij
;a
L
ij
2S;a
U
ij
2SÞ,
then all the preference values of the alternatives consists
the decision matrix e
A¼ð
~
a
ij
Þ
mn
and by Eq. (4) we can get
the ideal point ~
a
þ
¼ð
~
a
þ
1
;~
a
þ
2
;...;~
a
þ
n
Þof attribute values.
Step2: If the information about the attribute weights is com-
pletely unknown, we solve the model (M-2) to obtain
the optimal weight vector w
¼ðw
1
;w
2
;...;w
n
Þ
T
; If the
information about the weights is partly known, then we
solve the (M-4) to determine the attribute weights. and
then by Eq. (3), we obtain the overall values
~
z
i
ðw
Þði¼1;2;...;mÞof the alternatives x
i
(i=1,2,...,m).
Step3: Utilize the formula (1) to compare each ~
z
j
ðw
Þwith all
~
z
i
ðw
Þði¼1;2;...;mÞ, we get the possibility degree
p
ij
¼pð~
z
i
ðw
ÞP~
z
j
ðw
ÞÞ, and then construct a complemen-
tary matrix as P=(p
ij
)
mm
. Where p
ij
P0;p
ij
þp
ji
¼
1;p
ii
¼
1
2
;i;j¼1;2;...;n:
Step4: Utilize the Eq. (15) to get the sum in each line of the
matrix P=(p
ij
)
mm
. Then we rank the
~
z
i
ðw
Þði¼1;2;...;mÞin descending order in accordance
with the values of p
i
(i=1,2,...,m).
Step5: Rank all the alternatives x
i
(i=1,2,...,m) and select the
best one(s) in accordance with the ~
z
i
ðw
Þði¼1;2;...;mÞ.
Step6: End.
5. An illustrative example
Let us suppose to evaluate the technological innovation capabil-
ity of enterprises, There are four enterprises denoted as x
1
,x
2
,x
3
,x
4
to be evaluated. These attributes, which are critical for the selec-
tion of the best enterprise, are the following (adapted from [7]):
(1) u
1
: innovation input capacity (2) u
2
: innovation management
capacity (3) u
3
: innovation inclined. (4) u
4
: research and develop-
ment capabilities (5) u
5
: manufacturing capacity. (6) u
6
: marketing
ability.
The four possible alternatives are to be evaluated using the lin-
guistic term set
S¼fs
4
¼extremely poor;s
3
¼very poor;s
2
¼poor;
s
1
¼slightly poor;s
0
¼fair;s
1
¼slightly good;s
2
¼good;
s
3
¼very good;s
4
¼extremely goodg
and the decision maker give the uncertain linguistic decision matrix
as listed in Table 1.
To get the best alternative(s), the following steps are involved:
Step1: From Table 1, we can get the ideal solution of the attri-
bute values as follows:
~
a
þ
¼ð½s
2
;s
4
;½s
3
;s
4
;½s
1
;s
3
;½s
2
;s
4
;½s
3
;s
4
;½s
3
;s
4
Þ
Step2: We solve the formula Eq. (13), then we have:w
*
= (0.0687,
0.1458, 0.2292, 0.3437, 0.1266, 0.0859)
T
By Eq. (3),we
obtain the overall values ~
z
i
ðw
Þði¼1;2;3;4Þof the alter-
natives x
i
(i=1,2,3,4).
~
z
1
ðw
Þ¼½s
1:1169
;s
2:4726
;~
z
2
ðw
Þ¼½s
0:9635
;s
2:4071
;
~
z
3
ðw
Þ¼½s
1:3138
;s
3:3111
;~
z
4
ðw
Þ¼½s
1:0323
;s
2:7425
Step3: To rank these collective overall preference values
~
z
i
ðw
Þði¼1;2;3;4Þ, we first compare each ~
z
j
ðw
Þwith
all ~
z
i
ðw
Þði¼1;2;3;4Þby using formula (1), and develop
a complementary matrix:
P¼
0:50:5391 0:3456 0:4698
0:4609 0:50:3177 0:4359
0:6544 0:6823 0:50:6146
0:5302 0:5641 0:3854 0:5
2
6
6
6
43
7
7
7
5
Step4: Summing all elements in each line of the matrix P, we have
p
1
¼1:8545;p
2
¼1:7145;p
3
¼2:4513;p
4
¼1:9788
Then we rank the collective overall preference values
~
z
i
ðw
Þði¼1;2;3;4Þin descending order in accordance with
the values of p
i
(i= 1,2,3,4):
~
z
3
ðw
Þ>~
z
4
ðw
Þ>~
z
1
ðw
Þ>~
z
2
ðw
Þ
Step5: Rank all the alternatives x
i
(i= 1,2,3,4) and select the best
one(s) in accordance with the ~
z
i
ðw
Þði¼1;2;3;4Þ:
x
3
x
4
x
1
x
2
thus the best alternative is x
3
.
If the information about the attribute weights is partly un-
known, and the attribute weights information are follows:
0:06 6w
1
60:1;0:16w
2
60:2;0:26w
3
60:3;
w
4
P2w
2
;w
3
w
2
60:1
w
5
w
6
6w
4
w
3
;w
4
w
5
60:25;
w
6
P0:6w
5
;w
j
P0;j¼1;2;...;6;X
6
j¼1
w
j
¼1
Step 1: From Table 1, we can get the ideal solution of the attri-
bute values as follows:
~
a
þ
¼ð½s
2
;s
4
;½s
3
;s
4
;½s
1
;s
3
;½s
2
;s
4
;½s
3
;s
4
;½s
3
;s
4
Þ
Step 2: We utilize the model (M-4) to establish the following sin-
gle-objective programming model:
min k
1
þk
2
þk
3
þk
4
3w
1
þ2w
2
þ4w
3
þw
4
þ3w
5
þ2w
6
6k
1
5w
2
þ3w
4
þ2w
5
þ6w
6
6k
2
6w
1
þw
3
þ5w
5
6k
3
5w
1
þ2w
2
þ2w
3
þ2w
4
þ4w
6
6k
4
0:06 6w
1
60:1;0:16w
2
60:2;
0:26w
3
60:3;w
4
P2w
2
;w
3
w
2
60:1
w
5
w
6
6w
4
w
3
;w
4
w
5
60:25;
w
6
P0:6w
5
Table 1
Uncertain linguistic decision matrix e
A
x
i
u
1
u
2
u
3
u
4
u
5
u
6
x
1
[s
1
,s
2
][s
2
,s
3
][s
1
,s
1
][s
2
,s
3
][s
1
,s
3
][s
2
,s
3
]
x
2
[s
2
,s
4
][s
0
,s
2
][s
1
,s
3
][s
1
,s
2
][s
2
,s
3
][s
0
,s
1
]
x
3
[s
1
,s
1
][s
3
,s
4
][s
0
,s
3
][s
2
,s
4
][s
0
,s
2
][s
3
,s
4
]
x
4
[s
1
,s
2
][s
2
,s
3
][s
0
,s
2
][s
1
,s
3
][s
3
,s
4
][s
1
,s
2
]
840 Y. Xu, Q. Da / Knowledge-Based Systems 21 (2008) 837–841
w
j
P0;j¼1;2;...;6;X
6
j¼1
w
j
¼1
Solving this model, we get the weight vector of attributes:
w
¼ð0:06;0:172;0:272;0:344;0:094;0:057Þ
T
and k
1
= 2.35, k
2
= 2.42, k
3
= 1.1, k
4
= 2.1By Eq. (3), we obtain the
overall values ~
z
i
ðw
Þði¼1;2;3;4Þof the alternatives x
i
(i= 1,2,3, 4).
~
z
1
ðw
Þ¼½s
1:028
;s
2:393
;~
z
2
ðw
Þ¼½s
0:924
;s
2:427
;
~
z
3
ðw
Þ¼½s
1:315
;s
3:356
;~
z
4
ðw
Þ¼½s
0:967
;s
2:702
Step 3: To rank these collective overall preference values
~
z
i
ðw
Þði¼1;2;3;4Þ, we first compare each ~
z
j
ðw
Þwith
all ~
z
i
ðw
Þði¼1;2;3;4Þby using formula (1), and develop
a complementary matrix:
P¼
0:50:5122 0:3165 0:46
0:4878 0:50:3138 0:4509
0:6835 0:6862 0:50:6327
0:54 0:5491 0:3673 0:5
2
6
6
6
43
7
7
7
5
Step 4: Summing all elements in each line of the matrix P,we
have
p
1
¼1:7887;p
2
¼1:7525;p
3
¼2:5024;p
4
¼1:9564
Then we rank the collective overall preference values
~
z
i
ðw
Þði¼1;2;3;4Þin descending order in accordance with the val-
ues of p
i
(i= 1,2,3, 4):
~
z
3
ðw
Þ>~
z
4
ðw
Þ>~
z
1
ðw
Þ>~
z
2
ðw
Þ
Step 5: Rank all the alternatives x
i
(i= 1,2,3,4) and select the best
one(s) in accordance with the ~
z
i
ðw
Þði¼1;2;3;4Þ:
x
3
x
4
x
1
x
2
thus the best alternative is x
3
.
6. Concluding remarks
In the real world, sometimes the experts may estimate their
preferences with uncertain linguistic variables and construct
uncertain linguistic preference relations due to their vague knowl-
edge, environment or time pressure, or his/her limited expertise
about the problem domain, etc. In this paper, we have investigated
the multiple attribute decision making problems, in which the
attribute values take the form of uncertain linguistic variables,
and the information about attribute weights is completely un-
known. To determine the attribute weights, we have established
a simple optimization model based on the concept of deviation de-
gree of uncertain linguistic variables and the ideal point of uncer-
tain linguistic multiple attribute values. Solving the model, we
obtain a simple and exact formula for obtaining the attribute
weights. The prominent characteristic of the developed method
is that it can relieve the influence of subjectivity of the DMs and
utilize the decision information sufficiently. For the situations
where the information about the attribute weights is partly known,
we establish an optimization model to determine the weights, and
then we utilize the uncertain linguistic weighted average (ULWA)
operator to aggregate the uncertain linguistic variables corre-
sponding to each alternative. We utilize the formula of possibility
degree of uncertain linguistic variables to compare the aggregated
uncertain linguistic information, and then rank the alternatives. Fi-
nally, we give an example of practical application of the developed
method to evaluate the technological innovation capability of
enterprises.
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