Download The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables

The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu, Yu Su
The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu*, Yu Su
Information Management School , Shandong Economic University
Jinan Shandong 250014, China
[email protected](Corresponding Author:Peide liu)，[email protected]
doi: 10.4156/jcit.vol5.issue4.5
Abstract
The purpose of this paper is to extend the TOPSIS (Technique for Order Performance by Similarity
to Ideal Solution) method for solving multiple attribute decision making (MADM) problems with
trapezoid fuzzy linguistic variables. To begin with, this paper introduces the concept of the trapezoid
fuzzy linguistic variables, and defines the distance between two trapezoid fuzzy linguistic variables in
order to deal with trapezoid fuzzy linguistic variables directly. This paper proposed a combined weight
to highlight the importance of weights. Then, the combined weights of each attribute can be determined
by the maximizing deviation method and the non-linear weighted comprehensive method. Furthermore,
the relative closeness degree is defined based on the TOPSIS method to determine the ranking order of
all alternatives by calculating the distances to both the positive ideal solution and negative ideal
solution, respectively. Finally, an illustrative example is given to verify the feasibility and effectiveness
of this method.
Keywords: The Extended TOPSIS, The Trapezoid Fuzzy Linguistic Variables, Multiple Attribute
Decision Making (MADM), The Maximizing Deviation Methods, The Non-Linear Weighted
Comprehensive Method and The Combined Weight
1. Introduction
Multiple attribute decision making (MADM) under the linguistic environment is an interesting
research topic which has been receiving more and more attention in recent years [1-5]. TOPSIS (the
technique for order preference by similarity to ideal solution), developed by Hwang and Yoon [6], is
widely used in MADM. The basic concept of TOPSIS is that the best alternative which we chose from
all of the alternatives should be the least different from the positive-ideal solution and the most
different from the negative-ideal solution. The attribute values and weights often take the form of exact
values [7] or fuzzy numbers [8-11]. The attribute values, however, are only described by the linguistic
variables [12, 13] in real situation, because of time pressure, lack of knowledge, and their limited
attention and information processing capabilities. All the linguistic variables must be transformed into
fuzzy numbers in order to calculate the distance from the ideal solution. Therefore, this method fails to
deal with the MADM problems based on the linguistic variables directly. Therefore, we must extend
the TOPSIS in order to solve these problems.
In this paper, the extended TOPSIS has been proposed for solving the MADM problems based on
the trapezoid fuzzy linguistic variables. We can compute the distance between two trapezoid linguistic
variables directly based on the new distance formula.
To do so, the remainder of this paper is shown as follows: In section two, the concept of the
trapezoid fuzzy linguistic variables are briefly reviewed, and the distance between two trapezoid fuzzy
linguistic variables is defined to deal with the trapezoid fuzzy linguistic variables directly. In section
three, we extend the TOPSIS method. Firstly, the maximizing deviation method and the non-linear
weighted comprehensive method can be used to compute the combined weights of each attribute based
on the distance. Then, the relative closeness degree is defined to determine the ranking order of all
alternatives by calculating the distances from the positive ideal solution and the negative ideal solution,
respectively. In section four, the proposed method is illustrated with an example, and some conclusions
are pointed out in section five. In Appendix A, the proof of definition 2.3 is provided.
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Journal of Convergence Information Technology
Volume 5, Number 4, June 2010
2. Trapezoid Fuzzy Linguistic Variables
Suppose that S  s i |i  t ,   1,0,1,  , t  is a finite and totally ordered discrete term set,
where t is a non-negative integer and s i represents a possible value for a linguistic variable. For
example, the set S could be given as follows [14] where t equals four:
S  {s4  extremelypoor , s3  verypoor , s2  poor , s1  slightlypoor , s0  fair ,
s1  slightlygood , s2  good , s3  verygood , s4  extremlygood }
Usually, in these cases, si and s j must satisfy the following characteristics [14]:
(1) The set is ordered: si
 s j , iff i  j ;
(2) There is the negation operator: neg ( s i )  s i , especially, neg ( s 0 )  s 0 .
(3)Maximum operator: max( s i , s j )  s i , if i  j ;
(4)Minimum operator: min(s i , s j )
 s j , if i  j ;
In the process of information aggregation, however, some results may not exactly match any
linguistic labels in S . To preserve all the given information, Xu [14] extend the discrete term set S to
a continuous term set
S  {s | s q  s  sq , a  [  q , q ]} , where s
characteristics above and q (q
meets all the
 t ) is a sufficiently large positive integer. If s  S , then we call
s the original term, otherwise, we call s the virtual term. In general, the decision makers use
the original linguistic terms to evaluate alternatives, and the virtual linguistic terms can only appear in
operation.
Definition 2.1[15]: Let s , s be any two linguistic variables, then we defined the distance
between s and s  as:
d ( s , s  )    
(1)
In some situations, however, the decision makers may provide fuzzy linguistic information because
of time pressure, lack of knowledge, and their limited attention and information processing
capabilities. We defined the concept of trapezoid fuzzy linguistic variable as follows [15]:
39
The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu, Yu Su
Definition 2.2[15]


Let s  [ s , s , s , s ]  S , where s , s  , s , s  S , s  and s indicate
the interval in which the membership value is 1, with s and s indicating the lower and upper


values of s , respectively, then s is called a trapezoid fuzzy linguistic variable, which is
characterized by the following member function (see Figure 1.):
0

 d ( s , s )
 d (s , s )
  
 ） 1
（

S

 d ( s , s )
 d ( s , s )

0
s q  s  s
s  s  s 
s   s  s
s  s  s
s  s  s q

where S is the set of all trapezoid fuzzy linguistic variables. Especially, if any two of
 ,  ,  , are
 ,  ,  ,

equal, then s is reduced to a triangular fuzzy linguistic variable; if any three of

are equal, then s is reduced to an uncertain linguistic variable.

Figure 1. A trapezoid fuzzy linguistic variable s [16]
Consider
any
three
trapezoid
fuzzy
linguistic
variables

s  [s , s , s , s ] ,



s1  [s1 , s 1 , s 1 , s1 ] and s2  [ s 2 , s2 , s 2 , s2 ]  S , and suppose that   [0,1] and 1  [0,1] ,
then their operational laws are defined as follows:
 
(1) s1  s2  [ s1  2 , s1  2 , s1  2 , s1 2 ]

(2) s  [ s , s , s , s ]  [ s , s , s , s ]
   
(3) s1  s2  s2  s1



(4)(  1 ) s   s  1s
 


(5)（s  s1）  s   s1
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Journal of Convergence Information Technology
Volume 5, Number 4, June 2010
Definition
2.3

s1  [ s1 , s 1 , s 1 , s1 ]
Let
,

s2  [s2 , s2 , s 2 , s2 ]
and


s3  [s3 , s3 , s 3 , s3 ]  S be any three trapezoid linguistic variables, then we define the distance


s1 and s 2 as follows:
between
(1   2 ) 2  ( 1   2 ) 2  ( 1   2 )2  (1  2 )2
 
d ( s1 , s2 ) 
4 （2t）2
 
(2)


s 2 , and 2t is the number of linguistic terms
 


s1 from s 2 is. The
where d ( s1 , s2 ) is called the distance between s1 and
in the set S .
Obviously, the smaller the value of d ( s1 , s2 ) is, the closer the distance of
 
properties of the distance d ( s1 , s2 ) are shown as follows:
 
(1)0  d(s1, s2 ) 1;
 
 
(2)d(s1, s2 )  0, iff s1  s2 1  2，1  2，1   2，1 2;
 
 
(3)d(s1, s2 )  d(s2 , s1)
 
 
 
(4)d(s1, s2 )  d(s2 , s3 )  d(s1, s3 )
The proof of Definition 2.3 is shown in Appendix A.
3. The extended TOPSIS
A multiple attribute decision making problem under fuzzy linguistic environment is shown as
follows [16]:
Let X  {x1 , x 2 ,  , x n } be the set of alternatives, and U  {u1 , u 2 ,  , u m } be the set of
attributes.
Let
w  [ w1 , w2 ,  , wm ]T be the combined weight vector of attributes,
where wi  0 , i  1,2,  , m , and
m
vector of attributes, where

j 1
j
m
w
i 1
i
 1 . Let   {1 ,  2 , ,  m } be the objective weight
 1 , 0   j  1 and j  1,2, , m . Let   {1 ,  2 ,  ,  m }
be the subjective weight vector of attributes, given by the decision makers directly,
m
where

j 1
j


 1 , 0   j  1 , and j  1,2,  , m . Let A  (aij ) nm be the fuzzy linguistic
41
The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu, Yu Su

( )
( )
( )
( )

decision matrix, where a ij  [ a ij , a ij , aij , a ij ]  S is the attribute value which takes the form
of trapezoid fuzzy linguistic variables, given by the decision makers, for the alternative
xi  X ( i  1,2, , n ) with respect to the attribute u j  U ( j  1,2,  , m ). Let

 

a i  [a i1 , ai 2 , , aim ] be the vector of the attribute values with respect to the alternative xi ,
where i
 1,2, , n .
3.1 The method of determining the combined weights
We proposed combined weights that they can not only avoid the subjectivity from the decision
makers’ personal bias, but also confirm the objectivity, to highlight the importance of weights.
3.1.1 Computing the objective weights: the maximizing deviation method
The maximizing deviation method is proposed by Wang [17] to deal with the MADM problems with
numerical information. The maximizing deviation method is applied to calculate the weight of each
attribute when the attribute weights are unknown.
The deviation value is smaller, then the attribute should be assigned the smaller weight or vice versa.
The deviation value is represented by the distance between two linguistic variables.
For
the
attribute
u j  U ( j  1,2,  , m ), the deviation value of alternative
xi  X ( i  1,2,, n ) to all the other alternatives can be defined as:
n
 
Dij ( j )   d (aij , alj ) j ,
l 1
where
i  1,2,, n , j  1,2, , m .
Then D j ( j ) 
alternatives
where i
to
n
n
 
D
(

)

 ij j  d (aij , alj ) j represents the total deviation value of all
the
n
i 1
i 1 l 1
other
alternatives
for
the
attribute
u j U
(
j  1,2, , m ),
m
m n
n
 
 1,2, , n . D ( j )   D j ( j )   d (aij , alj ) j represents the total deviation
j 1
j 1 i 1 l 1
value of all attributes to all alternatives, where
j
represents the weight of the attribute

u j  U ( j  1,2, , m ) and aij represents the attribute value of the alternative xi  X
42
Journal of Convergence Information Technology
Volume 5, Number 4, June 2010
(i
 1,2, , n ) with respect to the attribute u j  U ( j  1,2, , m ).
The maximum deviation model based on the maximum deviation method can be constructed as
follows:
m n
n

 

max
(

)
D
d (aij , a lj ) j

j

j 1 i 1 l 1

m
 s.t.  2j  1,  j  0, j  1,2,  , m

j 1
(3)
where  j represents the weight of the attribute
 1,2, , n ) with respect to the attribute u j  U
attribute value of the alternative xi  X ( i
(

u j  U ( j  1,2, , m )and aij represents the
j  1,2,  , m ).
Constructed the Lagrange Function as:
m
m n
n
 
L( j ,  )   d (aij , a lj ) j   (  2j  1)
j 1 i 1 l 1
j 1
where  is the Lagrange multiplier. The partial differential of
L( j ,  ) with respect to  j and 
are:
n
n
 L( j ,  )
 

d (aij , alj )  2 j  0


  j
i 1 l 1
 L( ,  ) m
j

   2j  1  0
 
j 1
Therefore,  j and

are determined by

2  



j 



The
m
n
n
 
（
  d (aij , alj )）2
j 1 i 1 l 1
n
n
 
ij , a lj )
 d (a
i 1 l 1
m
n
n
 
（
  d (aij , alj )）2
j 1
i 1 l 1
 j can be normalized as:
43
The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu, Yu Su
n
n
 
ij , a lj )
 d (a
j 
m
i 1 l 1
n
n
(4)
 
 d (aij , alj )
j 1 i 1 l 1
The theoretic foundation of this method is based on information theory: the attribute providing more
information should be assigned the bigger weight.
3.1.2 Computing the combined weights: the non-linear weighted comprehensive method
Supposed that the vector of the subjective weight, given by the decision makers directly,
is   [1 ,  2 ,  ,  m ] , where
m

j 1
j
 1 , 0   j  1 and j  1,2, , m . The vector of the
objective weight is   [1 ,  2 ,  ,  m ] , where
m

j 1
j
 1 , 0   j  1 and j  1,2, , m .
Therefore, the vector of the combined weight W  [ w1 , w2 ,  , wm ] can be defined as:
wj 
j j
m
 (
j 1
m
where
w
j 1
j
j
(5)
j )
 1 , 0  w j  1 and j  1,2, , m .
We aggregate the objective weight and subjective weight by non-linear weighted comprehensive
method. The larger the value of the objective weight and subjective weight are, the larger the
combined weight is, based on the multiplier effect. On the contrary, the smaller the combined weight
is.
3.2 The extended TOPSIS
We use the extended TOPSIS method to ranking the alternatives based on the combined weight and
the distance between two trapezoid fuzzy linguistic variables we just defined. The decision steps are
shown as follows:


Step1: Construct the fuzzy linguistic decision matrix A  ( aij ) nm :
u1
u2

um
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Journal of Convergence Information Technology
Volume 5, Number 4, June 2010

 a 11

  a 21

A
 

 a n1


a 12

a 22


an2
( )

a 1m  x1

a 2 m  x 2
  
 
a nm  x n




( )
( )

( )
where a ij  [ aij , aij , a ij , aij ]  S is the attribute value of alternative xi  X ( i  1,2,  , n )
with respect to attribute
u j  U ( j  1,2,  , m ), which takes the form of trapezoid fuzzy linguistic
variable, given by the decision maker.
The normalization process is not necessary, due to the fact that all the attribute values are assessed
using the same set of linguistic variables. In other words, the fuzzy linguistic decision matrix equals the
normalized matrix.
If the combined weight is required in the calculation process, then the step 2 and step 3 can be
executed, otherwise, we can skip to the following two steps, and execute the step 4 directly.
Step 2: utilize the Eq.(4) to compute the objective weights:
m
where

j 1
j
  {1 ,  2 , ,  m }
,
 1 , 0   j  1 and j  1,2,  , m .
Step 3: utilize the Eq. (5) to compute the combined weights: W  [ w1 , w2 ,  , wm ]
Step 4: Construct the weighted fuzzy linguistic decision matrix:


Considering the different importance of each attribute, the weighted decision matrix R  ( rij ) nm
will be constructed as follows:

 r11

R   r21

 rn1
where

r12
r22

rn 2








r1m   w1a11 w2 a12  wm a1m 
r2 m    w1a21 w2 a22  wm a2 m 


 
   
rnm   w1an1 w2 an 2  wm anm 


rij  wj aij  [wj aij( ) , wj aij(  ) , wj aij( ) , w j aij( ) ] , and w j is the weight of the j th attribute, and
 
rij  S is the weighted attribute value of alternative xi  X ( i  1,2, , n ) with respect to attribute
u j  U ( j  1,2, , m ), which takes the form of trapezoid fuzzy linguistic variables.
Step 5: Determine the positive ideal solution X


and the negative ideal solution X .

The positive ideal solution X indicates the most preferable alternative, and the negative ideal
solution X

indicates the least preferable alternative. Both of them can be defined as:
45
The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu, Yu Su
  




 


X   {r1 , r2 , r3 , , rm } , where rk  [rk( ) , rk(  ) , rk( ) , rk( ) ] , rk( )  max{rij( ) } ,
i






rk(  )  max{rij(  ) } , rk( )  max{rij( ) } , rk( )  max{rij( ) } , i  1,2,  , n; k  1,2,  , m ;
i
i
i
  




 


X   {r1 , r2 , r3 , , rm } , where rk  [rk( ) , rk(  ) , rk( ) , rk( ) ] , rk( )  min{rij( ) } ,
i






rk(  )  min{rij(  ) } , rk( )  min{rij( ) } , rk( )  min{rij( ) } , i  1,2,  , n; k  1,2,  , m ;
i
i
i
Step 6: Utilize the Eq. (2) to calculate the distance
 

d (rij , r j ) between rij and

 


rj ( j  1,2,  , n ), and the distance d (rij , r j ) between rij and r j , j  1,2,  , n;
Step 7: Calculate the distance of each alternative from the positive ideal solution X

and the

negative ideal solution X .

The distance of each alternative from the positive ideal solution X can be currently calculated as:
S i 
k
 
 d (r , r
j 1
ij
j
) 2 , i  1,2, , m ,
and the distance of each alternative from the negative ideal solution X
S i 
k
 
 d (r , r
j 1
ij
j

can be currently calculated as:
) 2 , i  1,2,  , m
Step 8: Calculate the relative closeness degree to the ideal solution.
The closeness coefficient of the alternative xi  X ( i
 1,2, , n ) with respect to ideal solution
is defined as:
S i
,
C  
S i  S i

i
where
(6)
0  Ci  1 and i  1,2, , n .
Obviously, if an alternative xi  X (

from X , then
i  1,2, , n ) becomes closer to X  and farther
Ci approaches 1.
Step 9: Rank the alternatives.
A set of alternatives can now be ranked according to the descending order of
the maximum value of
Ci is the best.
46
Ci and the one with
Journal of Convergence Information Technology
Volume 5, Number 4, June 2010
4. The illustrative example
In this section, a decision making problem of assessing cars for buying [2, 16] is used to illustrate
the developed method.
There is a customer who intends to buy a car. Four types of cars xi ( i
 1,2,3,4 ) are available.
The customer takes into account four attributes to decide which car to buy: 1) G 1 : economy, 2) G 2 :
comfort, 3) G 3 : design, and 4) G 4 : safety. The decision maker evaluates these four types of cars xi
(
i  1,2,3,4 ) under the attributes
G i ( i  1,2,3,4 ) (whose weight vector is
  [0.3,0.2,0.1,0.4] giving by the decision makers) by using the linguistic scale
S  {s4  extremelypoor , s3  verypoor , s2  poor , s1  slightlypoor , s0  fair ,
s1  slightlygood , s2  good , s3  verygood , s4  extremlygood }
and gives a fuzzy linguistic decision matrix as listed in Table 1.

Table 1. Fuzzy linguistic decision matrix A
Gi
x1
x2
x3
x4
G1
[ s 3 , s  2 , s 0 , s1 ]
[ s  2 , s 0 , s1 , s 2 ]
[ s 1 , s1 , s 3 , s 4 ]
[ s 0 , s1 , s 2 , s 4 ]
G2
[ s 1 , s 0 , s 3 , s 4 ]
[ s 0 , s1 , s 2 , s 3 ]
[ s  4 , s 3 , s 1 , s1 ]
[ s 1 , s 2 , s 3 , s 4 ]
G3
[ s 0 , s1 , s 2 , s 4 ]
[ s 1 , s 0 , s 3 , s 4 ]
[ s1 , s 2 , s 3 , s 4 ]
[ s  2 , s 0 , s1 , s 2 ]
G4
[ s  2 , s 1 , s 0 , s 2 ]
[ s 1 , s 0 , s 2 , s 3 ]
[ s  2 , s 1 , s 0 , s1 ]
[ s1 , s 2 , s 3 , s 4 ]
In the following, we utilize the extended TOPSIS method to obtain the most desirable car.


Step 1: Construct the fuzzy linguistic decision matrix A  ( aij ) nm based on Table 1.
[s3 , s2 , s0 , s1 ] [s1 , s0 , s3 , s4 ] [s0 , s1 , s2 , s4 ] [s2 , s1 , s0 , s2 ]
  [s2 , s0 , s1 , s2 ] [s0 , s1 , s2 , s3 ] [s1 , s0 , s3 , s4 , ] [s1 , s0 , s2 , s3 ] 

A
 [s1 , s1 , s3 , s4 ] [s4 , s3 , s1 , s1 ] [s1 , s2 , s3 , s4 ] [s2 , s1 , s0 , s1 ]


 [s0 , s1 , s2 , s4 ] [s1 , s2 , s3 , s4 ] [s2 , s0 , s1 , s2 ] [s1 , s2 , s3 , s4 ] 
The objective weight is given by the decision maker, so we can skip to step 2 and step 3, and execute
the step 4 executed directly.
Step 4: Construct the weighted fuzzy linguistic decision matrix:
47
The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu, Yu Su
[s0.9 , s0.6 , s0 , s0.3 ] [s0.2 , s0 , s0.6 , s0.8 ]
  T  [s , s , s , s ] [s , s , s , s ]
0 0.2 0.4 0.6
R  A     0.6 0 0.3 0.6
[s0.3 , s0.3 , s0.9 , s1.2 ] [s0.8 , s0.6 , s0.2 , s0.2 ]
 [s0 , s0.3 , s0.6 , s1.2 ] [s0.2 , s0.4 , s0.6 , s0.8 ]
Step 5: Determine the positive ideal solution X

[s0 , s0.1, s0.2 , s0.4 ]
[s0.1, s0 , s0.3 , s0.4 ]
[s0.1, s0.2 , s0.3 , s0.4 ]
[s0.2 , s0 , s0.1, s0.2 ]
[s0.8 , s0.4 , s0 , s0.8 ]
[s0.4 , s0 , s0.8 , s1.2 ] 
[s0.8 , s0.4 , s0 , s0.4 ]
[s0.4 , s0.8 , s1.2 , s1.6 ] 

and the negative ideal solution X .
X   {[ s0 , s0.3 , s0.9 , s1.2 ],[ s0 , s0.4 , s0.6 , s0.8 ],[ s0.1 , s0.2 , s0.3 , s0.4 ],[ s0.4 , s0.8 , s1.2 , s1.6 ]}
X   {[ s0.9 , s0.6 , s0 , s0.3 ],[ s0.8 , s0.6 , s0.2 , s0.2 ],[ s0.2 , s0 , s0.1 , s0.2 ],[ s0.8 , s0.4 , s0 , s0.4 ]}
 

Step 6: Utilize the Eq. (2) to calculate the distance d ( rij , r j ) between rij and

 


3,4 ), and the distance d (rij , r j ) between rij and r j ( j  1,2，
3,4 ).
rj ( j  1,2，
 
 
 
 
d (r11 , r1 )  0.1125 ， d (r12 , r2 )  0.0820 ， d (r13 , r3 )  0.0108 , d (r14 , r4 )  0.1392
 
 
 
 
d (r21 , r1 )  0.0676 ， d (r22 , r2 )  0.0839 , d (r23 , r3 )  0.0177 , d (r24 , r4 )  0.0791
 
 
 
 
d (r31 , r1 )  0.0188 ， d (r32 , r2 )  0 , d (r33 , r3 )  0 , d (r34 , r4 )  0.15
 
 
d (r41 , r1 )  0.0188 ； d (r42 , r2 )  0.0960 ,
 
 
d (r43 , r3 )  0.0286 , d (r44 , r4 )  0
 
d (r11 , r1 )  0 ，
 
 
d (r13 , r3 )  0.0198 , d (r14 , r4 )  0.025
 
d (r12 , r2 )  0.0943 ,
 
 
 
 
d (r21 , r1 )  0.0496 ，d (r22 , r2 )  0.0964 , d (r23 , r3 )  0.0188 , d (r24 , r4 )  0.0791
 
 
d (r31 , r1 )  0.1044 d (r32 , r2 )  0 ,
 
 
d (r33 , r3 )  0.0286 , d (r34 , r4 )  0 ,
 
 
d (r41 , r1 )  0.1044 d (r42 , r2 )  0.1104 ,
 
 
d (r43 , r3 )  0 , d (r44 , r4 )  0.15
Step 7: Utilize the Eq. (2) to calculate the distance of each alternative from the positive ideal
solution X


and the negative ideal solution X .
S1  0.1972 , S 2  0.1348 ,
S 2  0.1355 , S 3  0.1082 ,
S 3  0.1512 , S 4  0.1020 ， S1  0.0995 ,
S 4  0.2135
Step 8: Utilize the Eq. (6) to calculate the closeness coefficient to the ideal solution.
C1  0.3354 , C 2  0.5013 , C 3  0.4171 , C 4  0.6767
Step 9: Rank the alternatives.
According to the closeness coefficient, the ranking order of all alternatives is x 4  x 2  x3  x1 .
48
Journal of Convergence Information Technology
Volume 5, Number 4, June 2010
Obviously, the best alternative is x 4 .
The order calculated by the subjective weights is the same as Xu’s[16], which proves the method
proposed in this paper is effective and feasible.
Then, we utilize the combined weight to construct the weighted matrix, and execute the approach.
We rank the alternatives and get the order is: x4  x2  x1  x3 . Obviously, the best alternative is x 4 .
The calculation process of the combined weight is shown as follows.
Step 2: Utilize the Eq. (4) to compute the objective weights:
2.69
3.3684
3.9704
 0.2115 ,  2 
 0.2648 ,  3 
 0.3121 ,
12.721
12.721
12.721
2.6922
4 
 0.2116
12.721
1 
then we get the vector of the subjective weights: 
 {0.2115，
0.2648，
0.3121，
0.2116}
Step 3: Utilize the Eq. (5) to compute the combined weights:
 
 
 
 
d ( a11 , a11 )  d ( a 21 , a 21 )  d ( a 31 , a 31 )  d ( a 41 , a 41 )  0 ,
 
 
d (a12 , a 22 )  d (a 22 , a12 )  0.125 ,
 
 
d (a11 , a 21 )  d (a 21 , a11 )  0.1654 ,
 
 
 
 
d (a12 , a12 )  d (a22 , a22 )  d (a32 , a32 )  d (a42 , a42 )  0
 
 
d (a11 , a31 )  d (a31 , a11 )  0.3480 ,
 
 
d (a12 , a 32 )  d (a32 , a12 )  0.4098
 
 
d (a11 , a 41 )  d (a 41 , a11 )  0.3480 ,
 
 
d (a12 , a 42 )  d (a 42 , a12 )  0.125
 
 
d (a 21 , a31 )  d (a31 , a 21 )  0.1976 ,
 
 
d (a 22 , a32 )  d (a32 , a 22 )  0.4193
 
 
d (a 21 , a 41 )  d (a 41 , a 21 )  0.1976 ,
 
 
d (a 22 , a 42 )  d (a 42 , a 22 )  0.125
 
 
d (a31 , a 41 )  d (a 41 , a31 )  0.0884
 
 
d (a32 , a 42 )  d (a 42 , a32 )  0.4801
 
 
 
 
d (a13 , a13 )  d (a23 , a23 )  d (a33 , a33 )  d (a43 , a43 )  0 ,
 
 
d (a14 , a 24 )  d (a 24 , a14 )  0.1654 ,
 
 
d (a13 , a 23 )  d (a 23 , a13 )  0.1083 ,
 
 
 
 
d (a14 , a14 )  d (a 24 , a 24 )  d (a34 , a34 )  d (a 44 , a 44 )  0
 
 
d (a13 , a33 )  d (a33 , a13 )  0.1083 ,
 
 
d (a14 , a34 )  d (a34 , a14 )  0.0625
49
The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu, Yu Su
 
 
d (a13 , a 43 )  d (a 43 , a13 )  0.1976 ,
 
 
d (a14 , a 44 )  d (a 44 , a14 )  0.3480
 
 
d (a 23 , a33 )  d (a33 , a 23 )  0.1768 ,
 
 
d (a 24 , a34 )  d (a34 , a 24 )  0.1976
 
 
d (a 23 , a 43 )  d (a 43 , a 23 )  0.1875 ,
 
 
d (a 24 , a 44 )  d (a 44 , a 24 )  0.1976
 
 
d (a33 , a 43 )  d (a 43 , a33 )  0.2864
 
 
d (a34 , a 44 )  d (a 44 , a34 )  0.375
The vector of the objective weights is: 
 [0.3,0.2,0.1,0.4] .
w1  0.27 , w2  0.23 , w3  0.13 w4  0.37 .
Then we get the vector of the combined weights is:
W  (0.27,0.23,0.13,0.37)
5. Conclusion
This paper investigated the multiple attribute decision making problems under fuzzy linguistic
environment. The extended TOPSIS method can deal with trapezoid fuzzy linguistic variables directly
according to the distance between two trapezoid fuzzy linguistic variables. This method can make the
computation process of the trapezoid fuzzy linguistic variables easily without loss of information and
make us obtain the feasible and effective result. The illustrative example shows that the method is
suitable for solving the MADM problems under fuzzy linguistic environment.
Appendix A:
1) Proof:
 t  1  t， t   2  t
 0   1   2  2t  0 
1   2
2t
 1   2
 1  0  
 2t
2

 1


Equally, we can conclude that:
 1   2
 0  
 2t
2
  2

  1 0   1
 2t



2
  2

  1 0   1
 2t



2

 1


2
2
2
2
 
 0  d ( s1 , s2 )  (1   2 )  ( 1   2 )  (21   2 )  (1  2 )  1
4 （2t）
2) Proof:
50
Journal of Convergence Information Technology
Volume 5, Number 4, June 2010
 
 s1  s2 , 1   2，1   2， 1   2，1   2
2
2
2
2
 
 d ( s1 , s2 )  (1   2 )  ( 1   2 )  (21   2 )  (1  2 )  0
4 （2t）
3) Proof:
(1   2 ) 2  ( 1   2 ) 2  ( 1   2 ) 2  (1   2 ) 2
 
 d ( s1 , s2 ) 
4 （2t）2

( 2  1 ) 2  (  2  1 ) 2  ( 2   1 )2  (2  1 ) 2
 
 d ( s2 , s1 )
2
4 （2t）
4) Proof:
 
 
 
 
 
 
d ( s1 , s2 ) + d ( s 2 , s3 )  d ( s1 , s3 )  [d ( s1 , s2 )  d ( s2 , s3 )]2  d 2 ( s1 , s3 )
 
 
 
 (d ( s1 , s2 )  d ( s2 , s3 )) 2  d 2 ( s1 , s3 )  0
 (1  2 )2  (1  2 )2  ( 1   2 )2  (1 2 )2  (2  3 )2  (2  3 )2  ( 2   3 )2  (2 3 )2 
2 [(1  2 )2  (1  2 )2  ( 1   2 )2  (1 2 )2 ]  [(2  3 )2  (2  3 )2  ( 2   3 )2  (2 3 )2 ] 
[(1  3 )2  (1  3 )2  ( 1   3 )2  (1 3 )2 ]  0
Suppose
A  (1  2 )2  (1  2 )2  ( 1   2 )2  (1 2 )2  (2  3 )2  (2  3 )2  ( 2   3 )2  (2 3 )2 
[(1  3 )2  (1  3 )2  ( 1   3 )2  (1 3 )2 ]
B  2 (1 2 )2  (1  2 )2  (1 2 )2  (1 2 )2   (2 3)2  (2  3)2  (2 3)2  (2 3)2
 A  (1  2 )2  (1  2 )2  (1   2 )2  (1 2 )2  (2  3 )2  (2  3 )2  ( 2   3 )2  (2 3 )2 
[(1  3 )2  (1  3 )2  (1   3 )2  (1 3 )2 ]  2(1  2 )(3  2 )  2(1  2 )(3  2 ) 
2( 1   2 )( 3   2 )  2(1 2 )(3 2 )
B  2 [(1 2 )2  (1  2 )2  (1   2 )2  (1 2 )2 ]  [(2 3 )2  (2  3 )2  ( 2   3 )2  (2 3 )2 ]
2
2
 2  (1 2 )(2 3 )  (1  2 )(2  3 )    (1   2 )( 2   3 )  (1 2 )(2 3 )   C
 2[ (1 2 )(2 3 )  (1  2 )(2  3 )  (1   2 )( 2   3 )  (1 2 )(2 3 ) ]
where
C  2[ (1   2 )( 2   3 )( 1   2 )( 2   3 )  ( 1   2 )(  2  3 )( 1   2 )( 2   3 ) 
(1   2 )( 2  3 )(1   2 )( 2   3 )  (1  2 )( 2  3 )( 1   2 )(  2  3 ) ]
51
The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables
Peide Liu, Yu Su
A B  0
 
 
 
 
 
 
 (d ( s1 , s2 )  d ( s2 , s3 )) 2  d 2 ( s1 , s3 )  0  (d ( s1 , s2 )  d ( s2 , s3 )) 2  d 2 ( s1 , s3 ) ;
 
 
 
 d ( s1 , s 2 )  0, d ( s 2 , s3 )  0, d ( s1 , s3 )  0
 
 
 
 d ( s1 , s2 )  d ( s2 , s3 )  d ( s1 , s3 ) .
 
 
 
d ( s1 , s 2 )  d ( s 2 , s3 )  d ( s1 , s3 )



Iff s1  s2  s3 , 1   2   3，1   2   3， 1   2   3，1   2  3 .
Then the proof of Definition 2.3 is completed.
6. Acknowledgment
This paper is supported by the Humanities and Social Sciences Research Project of Ministry of
Education of China(No.09YJA630088), and the Natural Science Foundation of Shandong Province
(No. ZR2009HL022). The authors also would like to express appreciation to the anonymous
reviewers for their very helpful comments on improving the paper.
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