an extension of topsis for fuzzy mcdm based on vague set theory

ISSN 1004-3756/05/1401/73 JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING CN11- 2983/N ©JSSSE 2005 Vol. 14, No. 1, pp73-84, March 30, 2005

AN EXTENSION OF TOPSIS FOR FUZZY MCDM BASED ON

VAGUE SET THEORY

Jue WANG San-Yang LIU Jie ZHANG

Department of Mathematics, Xidian University [email protected]

Abstract

This paper extends the TOPSIS to fuzzy MCDM based on vague set theory, where the characteristics of the alternatives are represented by vague sets. A novel score function is proposed in order to determine the vague positive-ideal solution (VPIS) and vague negative-ideal solution (VNIS). We present a weighted difference index to calculate the distance between vague values, by means of which the distance of alternatives to VPIS and VNIS can be calculated. Finally, the relative closeness values of various alternatives to the positive-ideal solution are ranked to determine the best alternative. An example is shown to illustrate the procedure of the proposed method at the end of this paper. Keywords: TOPSIS, fuzzy set, vague set, score function, fuzzy MCDM

1. Introduction The multicriteria decision-making problem

(Kickert 1978) is a kind of problem that all the alternatives in the choice set can be evaluated according to a number of criteria. A MCDM problem can be concisely expressed in matrix format as

1 2 nC C CL

1 11 12 1

2 21 22 2

1 2

n

n

m m m mn

A x x xA x x x

M

A x x x

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

L

L

M M

L

1 2{ , , , }nW w w w= L

where 1 2, , , mA A AL are possible

alternatives among which decision-makers have to choose, 1 2, , , nC C CL are the criteria with which alternative performance are measured,

ijx is the rating of alternative iA with respect to criteria jC , jw is the weight of criteria jC ,

1, 2, , ; 1, 2, ,i m j n= =L L .

In classical MCDM methods, TOPSIS method is one of the known methods as an alternative to the ELECTRE method, which was the firstly developed by Hwang and Yoon (Hwang and Yoon, 1981), and then was extended to other environment (Chen 2000; Lai et al., 1994; Liang 1999; Tsaur et al., 2002; Chu 2002). The basic concept of this method is that the selected best alternative should have the shortest distance from the positive-ideal solution

An Extension of TOPSIS for Fuzzy MCDM Based on Vague Set Theory

74 JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING / Vol. 14, No. 1, March, 2005

and the farthest distance from the negative-ideal solution.

Since the theory of fuzzy sets (Zadeh 1965) was proposed in 1965, it has been used for handling fuzzy decision-making problems (Chen 1988; Chen et al. 1989; Laarhoven and Pedrycz 1983; Yager 1978; Yager 1988). Gau et al. (1993) presented the concepts of vague set in 1993. They used a truth-membership function At and false-membership function Af to characterize the lower bound on the membership function

Aµ of a fuzzy set A. These lower bounds are used to create a subinterval on [0,1], namely [ ,1 ]A At f− , to generalize Aµ of fuzzy sets, where 1A A At fµ≤ ≤ − .

Recently, Chen (1994), Hong and Choi (2000) presented some techniques for handling fuzzy multicriteria decision-making problems based on vague set theory, where the characteristics of the alternatives are presented by vague sets. They defined intersection and union operators on vague values and used a score function to evaluate the degree of suitability to which an alternative satisfies the decision-maker’s requirement. In this paper, an extension of TOPSIS for fuzzy multicriteria decision-making problems based on vague set theory is given. Considering the special characters of the rating represented by vague values, the importance weight of each criterion may be evaluated by a three dimensions vector. A novel score function is developed to determine the vague positive-ideal solution (VPIS) and vague negative-ideal solution (VNIS). On the basis of the similarity measure (Chen 1997; Chen 1995; Hong and Kim 1999), a difference index between vague values is presented to calculate the distance to both VPIS

and VNIS. Finally, the separation distance and relative closeness of alternatives are defined to determine the ranking order of all alternatives. Furthermore, the decision-maker can select the best alternative.

This paper is organized as follows. Section 2 introduces the basic definitions of vague sets. On the basis of the similarity measure, we define a difference index between vague values in section 3. In section 4, we suggest a novel score function to evaluate the degree of suitability to the decision maker. The TOPSIS method for fuzzy MCDM based on vague set is presented in section 5. In section 6, an example is given to illustrate the proposed method. Finally, some conclusions are pointed out at the end of this paper.

2. Vague Sets Let U be the unverse of discourse,

1 2{ , , , }nU u u u= L , with a generic element of U denoted by iu . A vague set A in U is characerized by a truth-membership function At and a false-membership function Af ,

: [0,1]At U →

: [0,1]Af U →

where ( )A it u is a lower bound on the grade of membership of iu derived from the evidence for iu , ( )A if u is a lower bound on the negation of iu derived from the evidence against iu , and ( ) ( ) 1A i A it u f u+ ≤ . The grade of membership of iu in the vague set A is bounded to a subinterval [ ( ),1 ( )]A i A it u f u− of [0,1] . The vague value [ ( ),1 ( )]A i A it u f u− indicates that the exact grade of membership

( )A iuµ of iu may be unknown. But it is

WANG, LIU and ZHANG

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING / Vol. 14, No. 1, March, 2005 75

bounded by ( ) ( ) 1 ( )A i A i A it u u f uµ≤ ≤ − , where ( ) ( ) 1A i A it u f u+ ≤ . When the universe of discourse U is

continuous, a vague set A can be written as

[ ( ),1 ( )] / ,A i A i i iUA t u f u u u U= − ∈∫ (1)

When the universe of discourse U is discrete, a vague set A can be written as

1[ ( ),1 ( )] / ,

n

A i A i i ii

A t u f u u u U=

= − ∈∑ (2)

Definitions 2.1 Let x be a vague value, [ ,1 ]x xx t f= − , where [0,1]xt ∈ , [0,1]xf ∈ ,

and 0 1 1x xt f≤ ≤ − ≤ . The vague value can be divided into three parts: the truth-membership part (i.e., xt ), and the false-membership part (i.e., xf ), and the unknown part (i.e., 1 x xt f− − ). Definition 2.2 Let x be a vague value, where

[ ,1 ]x xx t f= − . If xt =1 and 0xf = (i.e., [1,1]x = ), then x is called a unit vague value.

If xt =0 and 1xf = (i.e., [0,1]x = ), then x is called a zeros vague value. Definition 2.3 Let x and y be two vague values, where [ ,1 ]x xx t f= − and

[ ,1 ]y yy t f= − . If x yt t= and x yf f= , then the vague values x and y are called equal (i.e., [ ,1 ] [ ,1 ]x x y yt f t f− = − ). Definition 2.4 Let A be a vague set of the universe of discourse 1 2{ , , , }nU u u u= L , where

1 1 1 2

2 2

[ ( ),1 ( )] / [ ( ),1( )] / [ ( ),1 ( )] /

A A A

A A n A n n

A t u f u u t uf u u t u f u u

= − + −+ + −L

If i∀ , ( ) 1A it u = and ( ) 0A if u = , then A is called a unit vague set, where 1 i n≤ ≤ . If i∀ ,

( ) 0A it u = and ( ) 1A if u = , then A is called a zero vague set, where 1 i n≤ ≤ .

Definition 2.5 Let A and B be two vague sets of the universe of discourse U ,

1 2{ , , , }nU u u u= L , where

1 1 1 2

2 2

[ ( ),1 ( )] / [ ( ),1 ( )] / [ ( ),1 ( )] /

A A A

A A n A n n

A t u f u u t uf u u t u f u u

= − + −+ −L

1 1 1 2

2 2

[ ( ),1 ( )] / [ ( ),1 ( )] / [ ( ),1 ( )] /

B B B

B B n B n n

B t u f u u t uf u u t u f u u

= − + −+ −L

If i∀ ,

1 1 1 1[ ( ),1 ( )] [ ( ),1 ( )]A A B Bt u f u t u f u− = − , then the vague sets A and B are called equal, where 1 i n≤ ≤ .

3. Difference Index between Vague Values

Measures of similarity between vague values and sets have been studied in many Literatures (Chen 1997; Chen 1995; Hong 1999). In 1997, Chen proposed a similarity measure CM between the vague values x and y as follows,

| ( ) ( ) |( , ) 1

2x y x y

Ct t f f

M x y− − −

= − (3)

He also defined a weighted similarity measure ( , )w

CM x y between the vague values x and y ,

1( , ) 1 | ( ) (

) ( ( )) |

wC x y x

y y y x x

M x y a t t b fb af c t f t f

= − ∗ − + ∗−

− + ∗ − − − (4)

where 0a c b≥ ≥ ≥ . However, Hong and Kim (1999) showed by

examples that the similarity measures proposed by Chen do not fit well in some cases, and presented a set of modified measure between vague values. In this paper, we only give the definition of modified weighted similarity



measure proposed by Hong and Kim, which is used to define a difference index. Definition 3.1 Let x and y be two vague values, [ ,1 ]x xx t f= − , [ ,1 ]y yy t f= − , where 0 1 1x xt f≤ ≤ − ≤ and 0 1 1y yt f≤ ≤ − ≤ , the weighted similarity measure between the vague values x and y can be evaluated by the function wM ,

(

)

1( , ) 1 | |

| |

| ( ) |

w x y

x y

y y x x

M x y a t ta b c

b f f c

t f t f

= − ∗ −+ +

+ ∗ − + ∗

+ − +

(5)

where a , b and c represent the weight of the truth-membership part, the weight of the false-membership part, and the weight of the unknown part of the vague values, respectively, a , b , c 0≥ .

Obviously, ( , ) [0,1]wM x y ∈ . The larger the value of wM , the more the similarity between the vague values x and y .

Based on the above definition of similarity measure, we define a difference index between two vague values in the following. Definition 3.2. Let x and y be two vague values, [ ,1 ]x xx t f= − , [ ,1 ]y yy t f= − , where 0 1 1x xt f≤ ≤ − ≤ and 0 1 1y yt f≤ ≤ − ≤ , the weighted difference index between the vague values x and y can be evaluated by the function wd ,

(

)

1( , ) | |

| |

| ( ) |

w x y

x y

y y x x

d x y a t ta b c

b f f c

t f t f

= ∗ −+ ++ ∗ − + ∗

+ − +

(6)

where a , b and c represent the weight of the truth-membership part, the weight of the false-membership part, and the weight of the unknown part of the vague values, respectively,

a, b , 0c ≥ . The larger the value of ( , )wd x y , the less the similarity between the vague values x and y .

Let x , y and z be three vague values, [ ,1 ]x xx t f= − , [ ,1 ]y yy t f= − , [ ,1 ]z zz t f= − .

Some important properties of wd are described as follows: Property 1 ( , ) 0wd x y x y= ⇔ = Property 2 ( , ) ( , )w wd x y d y x= Property 3 ( , ) ( , ) ( , )w w wd x y d y z d x z+ ≥ Proof. By applying (4), we can get

(

)

(

1( , ) ( , )

| | | |

| ( ) |

| | | |

| ( ) |

1

( | | | |

w w

x y x y

y y x x

x z y z

z z y y

x y y z

d x y d y za b c

a t t b f f

c t f t f

a t t b f f

c t f t f

a b ca t t t t

+ = ⋅+ +

∗ − + ∗ − +

∗ + − + +

∗ − + ∗ − +

∗ + − +

= ⋅+ +∗ − + − +

)

()

(| | | |)

(| ( ) |

| ( ) |)

1

| | | |

| ( ) | ( , )

x y y z

y y x x

z z y y

x z x z

z z x x

w

b f f f f

c t f t f

t f t f

a b c a t t b f f

c t f t fd x y

∗ − + −

∗ + − + +

+ − +

≥ ⋅+ +∗ − + ∗ − +

∗ + − += ■

Example 3.1 Let x and y be two vague values, [0.2,0.6]x = , [0.3,0.6]y = and 1a = ,

1b = and 0c = , the difference index between x and y is calculated as

(1( , ) 1 | 0.2 0.3 |1 1 0

1 | 0.4 0.4 |

wd x y = ∗ − ++ +∗ − +

WANG, LIU and ZHANG


) 0 | 0.3 0.4 (0.2 0.4) | 0.05

∗ + − −=

If 2a = , 1b = and 0c = , the difference index between x and y is calculated as

(

)

1( , ) 2 | 0.2 0.3 |2 1 0

1 | 0.4 0.4 | 0 | 0.3 0.4 (0.2 0.4) | 0.067

wd x y = ∗ − ++ +∗ − +∗ + − −

=

4. A Novel Score Function A novel score function is introduced in this

section in order to determine the vague positive-ideal solution (VPIS) and vague negative-ideal solution (VNIS).

Chen (1995), Hong and Choi (1999)

proposed the score function C and H respectively, and x xC t f= − , x xH t f= + . In some cases, however, these functions do not give sufficient information about alternatives because they only consider the truth-membership part (i.e., xt ) and the false-membership part (i.e., xf ), without the unknown part (i.e., 1 x xt f− − ). In this section, we suggest a new score function S to evaluate the degree of suitability that an alternative satisfies some criteria. The proposed function emphasizes on three parts, i.e. xt , xf and 1 x xt f− − simultaneously. Therefore, it provides a more useful way than those of Chen and Hong to help the decision maker make his decision. Definition 4.1 Let [ ,1 ]x xx t f= − be a vague value, where [0,1]xt ∈ , [0,1]xf ∈ , 1x xt f+ ≤ . The score of x can be evaluated by the score function S shown as follows:

1( )

2x x

x xt f

S x t f− −

= − −

3 1

2x xt f− −

= (7)

The larger the value of ( )S x , the more the degree of suitability that alternative satisfies some criteria. Example 4.1 Let x and y be the characteristics of alternative 1A and 2A with respect to a fixed criteria, respectively, and

[0,1]x = , [0,1]y = . By (7) we have 0 1 0( ) 0.05

2S x − −= = −

1.5 1 0.5( ) 02

S y − − =

Therefore, 2A satisfies the criteria with more degree of suitability than 1A . According to the function x xC t f= − , however, we can not distinguish them because ( ) ( ) 0C x C y= = . Example 4.2 Let x and y be the characteristics of alternative 1A and 2A with respect to a fixed criteria, respectively, and

[0.4,0.9]x = , [0.1,0.6]y = . Conditioned on (7), we can get

1.2 1 0.1( ) 0.52

S x − −= =

0.3 1 0.4( ) 0.552

S y − −= = −

Obviously, 1A satisfies the criteria with more degree of suitability than 2A . We can not know which one is better using the function H because ( ) ( ) 0.5H x H y= = . Certainly, there exist some vague values that the function S cannot distinguish, such as [0.2,0.6] and [0.1,0.9] . But this is an interesting case showed in theorem 4.9.

The score function S preserves some important properties as follows: Theorem 4.1 Let x be a vague value,



1 2[ , ]x x x= , and S be the score function, the ( ) 1S x = if and only if 1 1x = , 2 1x = .

Theorem 4.2 Let x be a vague value, [ ,1 ]x xx t f= − , and S be the score function,

then 1 ( ) 1S x− ≤ ≤ . Proof. For any vague values x , [ ,1 ]x xx t f= − , where 0 1xt≤ ≤ , 0 1xf≤ ≤ . Using (5), we have

3 1 3 1 1 0( ) 12 2

x xt fS x

− − × − −= ≤ =

3 1 3 0 1 1( ) 12 2

x xt fS x

− − × − −= ≥ = −

This implies 1 ( ) 1S x− ≤ ≤ . ■ Theorem 4.4 Let x and y be two vague values, 1 2[ , ]x x x= , 1 2[ , ]y y y= , then

( ) ( )S x S y≥ if and only if

1 1 2 2( ) / 3x y y x− ≥ − . Proof. By applying (5), we can get

1 23 1 1( )

2x x

S x− + −

=

1 23 1 1( )

2y y

S y− + −

=

If ( ) ( )S x S y≥ , it is easily seen that

1 2 1 23 3x x y y+ ≥ + . ■ Theorem 4.5 Let x and y be two vague values, 1 2[ , ]x x x= , 1 2[ , ]y y y= , x y≠ , and

( ) ( ).C x C y= If ( ) ( ),H x H y< then ( ) ( );S x S y< if ( ) ( ),H x H y> then ( ) ( )S x S y> .

Proof. When ( ) ( )C x C y= , we have ( ) ( ).H x H y≠ Otherwise, 1 2 1 2x x y y+ = +

and 1 2 1 2x x y y− = − , we obtain 1 1x y= , and

2 2x y= . Obviously, it is not in agreement with the assumption x y≠ .

Conditioned on (5), we have 1 23 1 1 2 ( ) ( ) 1( )

2 2x x C x H xS x− + − + −= =

1 23 1 1 2 ( ) ( ) 1( )2 2

y y C y H yS y− + − + −= =

Furthernore, we obtain

2( ( ) ( )) ( ) ( )( ) ( )2

C x C y H x H yS x S y − + −− =

Since ( ) ( )C x C y= , ( ) ( )( ) ( )

2H x H yS x S y −− =

This omplies that the theoren has been proved. ■ The theorem above shows that the function

S agrees with the function H when the function C is invalid . Let 1 2{ , , , }tA A A A= L be a set of alternatives to be ranked. If

1 2( ) ( ) ( )tC A C A C A= = =L , then the ranking order of all the alternatives using the function S is the same as the function H . Theorem 4.6 Let x and y be two vague values, 1 2[ , ]x x x= , 1 2[ , ]y y y= , x y≠ , and

( ) ( ).H x H y= If ( ) ( ),C x C y< then ( ) ( );S x S y< if ( ) ( ),C x C y> then ( ) ( )S x S y> .

Proof. By following the same scheme developed in theorem 4.6, we can get

2( ( ) ( )) ( ) ( )( ) ( )2

C x C y H x H yS x S y − + −− =

Since ( ) ( )H x H y= , ( ) ( ) ( ) ( )S x S y C x C y− = − . This imples that the theorem has been proved. ■

Theorem 4.7 shows that the function S agrees with the function C when the function H is invalid. Let 1 2{ , , , }sA A A A= L be a set of alternatives to be ranked. If

1 2( ) ( ) ( )sH A H A H A= = =L , then the ranking order of all the alternatives using the function S is the same as the function C . Theorem 4.7 Let Let x and y be two vague values, 1 2[ , ]x x x= , 1 2[ , ]y y y= , x y≠ . If

( ) ( )C x C y> , ( ) ( )H x H y> , then ( ) ( )S x S y> ; If ( ) ( ),C x C y< ( ) ( ),H x H y< then

( ) ( )S x S y< . This theorem is easy to be proved by

WANG, LIU and ZHANG


following the same manipulation above. Theorem 4.8 Let x and y be two vague values, 1 2[ , ]x x x= , 1 2[ , ]y y y= , x y≠ . If

( ) ( ),S x S y> then either ( ) ( ),C x C y> ( ) ( )H x H y< or ( ) ( )C x C y< , ( ) ( )H x H y> .

Proof. According to the definition of the function S , if ( ) ( )S x S y= , then

1 1 2 23 ( )x y y x∗ − = − (8)

and 1 1x y≠ . Otherwise, we obtain 2 2x y= . Obviously, this is not in agreement with the assumption x y≠ . Using the definition of C and H , we have

( )( )( )( )1 1 2 2 1 1 2 2

2 21 1 2 2

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

C x C y H x H y

x y y x x y y x

x y y x

− −

= − − − − + −

= − − − (9)

By substituting (8) into (9), we can get

( )( )2

1 1

( ) ( ) ( ) ( )

8( ) 0

C x C y H x H y

x y

− −

= − − <

This yields either ( ) ( )C x C y< , ( ) ( )H x H y> or ( ) ( )C x C y> , ( ) ( )H x H y< . ■

This theorem shows that inconsistent results appear using the function C and H respectively when the function S cannot distinguish the vague values. Let

1 2{ , , , }tA A A A= L be a set of alternatives to be ranked. If the function S cannot distinguish all the alternatives i.e., 1 2( ) ( ) ( )tS A S A S A= = =L , and the ranking order using the function C is

1 2 tA A Af fLf , then the ranking order by means of H is 1 1t tA A A−f fLf , which is opposite to the former, and vice versa. It is easy to say no better function can be selected from C and H when our function ( )S x is invalid.

After the degrees of suitability that all

alternatives are measured, we can construct a suitability matrix SM for m alternatives and n criteria in the fuzzy multicriteria decision-making based on vague set as follows:

( )ijSM s m n= × (10)

where ([ ,1 ])ij ij ijs S t f= − .

Based on the matrix SM , we can obtain the ranking order of m alternatives with respect to each criteria by ranking each column of the matrix SM for the purpose of determining the ideal solution in next section.

5. TOPSIS Method for Fuzzy MCDM based on Vague Set

An extension of the TOPSIS method for fuzzy MCDM based on vague set is given in the section.

In this paper, the importance weight of each criteria may be evaluated by a vector ( , ,i i ia b c ), 1 i n≤ ≤ , where ia , ib and ic represent the weight of the truth-membership part, the weight of the false-membership part, and the weight of the unknown part of the vague values with respect to the i -th criteria, respectively, and

, , 0i i ia b c ≥ .

Usually, a fuzzy MCDM problem based on vague set may be expressed concisely in a matrix format as

11 11 1 1

1 1

[ ,1 ] [ ,1 ]

[ ,1 ] [ ,1 [

n n

m m mn mn

t f t fM

t f t f

− −⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟− −⎝ ⎠

L

M

L

1 2[ , , , ]nW w w w= L

where ( , , )i i i iw a b c= , 1 i n≤ ≤ .

Based on the matrix SM , we define the



vague positive-ideal solution A∗ and vague negative-ideal solution A− , respectively, and

* * *1 2( , , , )nA r r r∗ = % % %L (11)

1 2( , , , )nA r r r− − − −= % % %L (12)

where *1 arg max( )i ijr s=% , 1 arg min( )i ijr s− =%

and ([ ,1 ]) ,ij ij ijs S t f SM= − ∈ 1 ,j n≤ ≤

1 i m≤ ≤ . The distance of each alternative from *A

and A− can be calculated as

* *

1([ ,1 ], ), 1, 2, ,

n

wi w ij ij jj

d d t f r i m=

= − =∑ % L (13)

1([ ,1 ], ), 1, 2, ,

n

wi w ij ij jj

d d t f r i m− −

== − =∑ % L (14)

Where ( )wd ⋅ is the weighted difference index between two vague values.

The relative closeness to the ideal solution is defined to determine the ranking order of all the alternatives once *

wid and wid − has been calculated. The relative closeness of each alternative is calculated as

* , 1,2, ,wii

wi wi

dR i m

d d

−

−= =+

L (15)

where 0 1iR≤ ≤ .

It is noted that only when the larger the value of wid − , the less similarity between each alternative and A− . Hence, the larger the value of iR , the better the i -th alternative.

Therefore, we may determine the ranking order of all the alternatives according to iR ,

1,2, ,i m= L and select the best alternative. In sum, the algorithm (TOPSIS-vague) for

fuzzy multicriteria decision-making based on vague set is given as follows.

Input: the rating for alternatives and the importance weight of the criteria W ;

Output: the ranking order of all alternatives. Step1 Choose the appropriate vector for the

importance weight of the criteria and the rating represented by vague value for alternatives with respect to criteria.

Step2 Construct the vague decision matrix M .

Step3 Calculate the degree of suitability that all alternatives satisfy each criteria using the function ( )S x

Step4 Construct the suitability matrix SM . Step5 Determine the vague positive-ideal

solution *A and vague negative-ideal solution A− by means of the matrix SM .

Step6 Calculate the separation distances to both vague positive-ideal solution *A and vague negative-ideal solution A− .

Step7 Calculate the relative closeness to the ideal solution of each alternative.

Step8 Determine the ranking order of all alternatives according to the relative closeness and select the best one.

6. Numerical Example The vague version of the TOPSIS method is

best illustrated in the following numerical example. Suppose that a high technology company needs to hire an engineer. Now there are eight candidates 1 2 8{ , , , }A A A A= L and six benefit criteria are considered:

(1) personality ( 1C ) (2) past experience ( 2C ) (3) educational level ( 3C ) (4) self-confidence ( 4C ) (5) emotional steadiness ( 5C ) (6) oral communication skill ( 6C )

WANG, LIU and ZHANG


The TOPSIS method is now applied to solve this problem and the procedure is summarized as follows: Step 1 The decision-maker evaluate the rating of alternatives and the importance weight with. respect to each criteria by using vague values (shown in Table 1) and the importance weights

(shown in Table 2), respectively. Step 2 Construct the vague decision matrix M . Step 3 Calculate the degree of suitability that eight alternatives satisfy each criteria by the score function S . Step 4 Construct the suitability matrix SM as follows:

Table 1 The ratings of eight candidates under six criteria

C1 C2 C3 C4 C5 C6

A1 [0.2, 0.4] [0.5, 0.8] [0.1, 0.6] [0.2, 1.0] [0.5, 0.7] [0.5, 0.6]

A2 [0.3, 1.0] [0.3, 0.4] [0.5, 0.9] [0.3, 0.7] [0.2, 1.0] [0.1, 1.0]

A3 [0.6, 0.8] [0.7, 0.8] [0.5, 0.6] [0.1, 0.4] [0.7, 0.8] [0, 0.8]

A4 [0.4, 0.9] [0.1, 0.9] [0.5, 0.8] [0.5, 0.6] [0.3, 0.4] [0.2, 0.3]

A5 [0.8, 0.9] [0.2, 0.5] [0.4, 0.6] [0.6, 0.7] [0.4, 0.8] [0.4, 0.9]

A6 [0.3, 0.7] [0.4, 0.6] [0.2, 0.4] [0.2, 0.1] [0.5, 0.6] [0.1, 0.7]

A7 [0.5, 0.6] [0.6, 0.7] [0.3, 0.7] [0.4, 0.9] [0.1, 0.9] [0.1, 0.6]

A8 [0, 0] [0.5, 0.9] [0.3, 0.6] [0.7, 0.8] [0, 0.9] [0, 0.2]

Table 2 The importance weight of the criteria

C1 C2 C3 C4 C5 C6

w (1, 1, 0) (2, 1, 0) (1.5, 0.5, 1) (3, 1.5, 1) (2, 1, 1) (2.5, 2, 1.5)

Table 3 The distance measurement

A1 A2 A3 A4 A5 A6 A7 A8

d+ 1.81 1.73 1.05 1.53 0.91 1.74 1.22 1.83

d- 1.56 1.75 1.77 1.88 1.99 1.43 1.85 1.22

Table 4 The relative closeness to ideal solution

A1 A2 A3 A4 A5 A6 A7 A8

R 0.4624 0.5025 0.6277 0.5518 0.6871 0.4521 0.6026 0.4012



.5 .5 .55 .2 .1 .05.05 .05 .2 .2 .2 .35.3 .45 .05 .65 .45 .6

.05 .4 .15 .05 .35 .55

.65 .45 .1 .25 0 .05.2 .05 .5 .2 .05 .5

.05 .2 .2 .05 .2 .551 .25 .25 .45 .1 .6

SM

− −⎛ ⎞⎜ ⎟− − − − −⎜ ⎟⎜ ⎟− − −⎜ ⎟

− − −⎜ ⎟= ⎜ ⎟− −⎜ ⎟− − − − −⎜ ⎟

⎜ ⎟− −⎜ ⎟⎜ ⎟− − − − −⎝ ⎠

Therefore, we can obtain the ranking order of all alternatives with respect to each criterion. Step 5 Determine the vague positive-ideal solution *A and vague negative-ideal solution A− :

* ([0.8, 0.9], [0.7, 0.8], [0.5, 0.9], [0.7, 0.8], [0.7, 0.8], [0.4, 0.9])A =

([0.0, 0.0], [0.2, 0.5], [0.1, 0.6], [0.1, 0.4],[0.3, 0.4], [0.0, 0.8])A− =

Step 6 Calculate the separation distances (shown in Table 3) between each alternative and the vague positive-ideal solution *A and vague negative-ideal solution A− , respectively. Step 7 Calculate the relative closeness to the ideal solution of each alternative (shown as Table 4). Step 8 The ranking order of all alternatives can be determined according to the relative closeness as follows:

5 3 7 4 2 1 6 8A A A A A A A Af f f f f f f Therefore, the best selection is candidate 5A .

7. Conclusions In this paper, we have extended TOPSIS

method to fuzzy MCDM based on vague set, where the characteristics of the alternatives were presented by vague values. We have defined a

difference index between two vague values and a novel score function to determine the vague positive-ideal solution and vague negative-ideal solution. Then an algorithm for fuzzy MCDM based on vague set has been given. The proposed techniques can efficiently help the decision-maker to make his decisions. However, we determined the values of weights only in terms of the preferences of decision-maker. We will find effective approach to determine the weights in future work. Also, applications of the proposed method will be required, such as conflict analysis and voting problems, and so on.

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WANG, LIU and ZHANG

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Jue Wang received the B. Sc degree in mathematics from Qingdao University, 2000. From August 2000, she has been pursuing his Doctor degree in Xidian University, Xi’an, China. Her research interests include optimization theory and applications, intelligent information processing, rough sets theory and application in data mining, signal processing, etc. She has published over ten journal and conference papers. San-Yang Liu was born in Xi’an, China, in 1959. From 1985 to 1989, he was with the department of mathematics of Xi’an Jiaotong University, where he received the Ph. D. degree in Mathematics. He is currently a professor and Doctor advisor at the Xidian University. His interests include optimization theory and applications.



Jie Zhang was born in Huaibei, Anhui province, P. R. China., in 1975. He received his B. Sc. in Applied Mathematics, in 1999 from Anhui Normal University and M. Sc. in Operation Research, in 2002 from Xidian University. From 2002, he has been pursuing

his Doctor Degree in the National Key laboratory for Radar Signal Processing, Xidian Univ. His research interests lie in the area of signal and information processing with application to wireless communications.

an extension of topsis for fuzzy mcdm based on vague set theory

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