1 INTRODUCTIONThe theory of fuzzy set [1] proposed by Zadeh (1965) had been widely applied in the field of mechanical design and mechanical manufacture. As a generalization of fuzzy set, Atanassov [2] introduced the concept of intuitionistic fuzzy set (IFS), and Gau and Buehrer [3] introduced the concept of vague set (VS). Bustince and Burillo [4] pointed out that the notions of IFS and VS were the same. With the help of truth-membership function and false-membership function, VS can describe the objective world more realistically and accurately compared to fuzzy sets. In recent years, vague set theory has been applied to many different fields, such as decision making [5~ 7], pattern recognition [8], mechanical design [9], etc. The distance measure between vague Sets was the foundation of the applications in these areas. The accuracy of distance measures will directly influence the results of decision making problems and the effectiveness of practical applications. This paper proposed TOPSIS based on vague sets and applied it to solve decision making problems in scheme evaluation of mechanical design.

2 BASIC NOTIONS OF VAGUE SETS MS Word Authors: please try to use the paragraph styles contained in this document. Definition 1 Let X be a space of points (objects), with a generic element of X denoted by x .A vague set A in X ischaracterized by a truth-membership function At and a

false-membership function Af . ( )At x is a lower bound on the grade of membership of x derived from the evidence for x , and ( )Af x is a lower bound on the negation of x derived from the evidence against x .

( )At x and ( )Af x are both associated a real number in the

interval [0, 1] with each point in X , where ( )At x ( ) 1Af x+ ≤ . That is

At [ ]: 0,1X → ,

Af [ ]: 0,1X → .This approach bounds the grade of membership of ix in the

vague set A to a subinterval ( ) ( )[ ,1 ]A At x f x− of[ ]0,1 .

The uncertainty degree (hesitation degree) of x to A can be evaluated by the uncertainty function ( )A xπ :

( ) 1A xπ = − ( )At x ( )Af x− ,

where ( ) [ ]0,1A xπ ∈ .

If ( )At x ( ) 1Af x+ = , i.e. ( ) 0A xπ = , x X∀ ∈ , Areverts back to a fuzzy set. Definition 2 If x and y are two vague values,

x = ( ) ( )[ ,1 ]A At x f x− , ( ) ( )[ ,1 ]A Ay t y f y= − .

x y≤ ⇔ ( ) ( )A At x t y≤ and ( ) ( )A Af x f y≥ ;

x y= ⇔ ( ) ( )A At x t y= and ( ) ( )A Af x f y= .

If A and B are two vague sets in X , then A B⊆⇔ ( ) ( )A Bt x t x≤ and ( ) ( )A Bf x f x≥ , x X∀ ∈ ;

A B=⇔ ( ) ( )A Bt x t x= and ( ) ( )A Bf x f x= , x X∀ ∈ .

TOPSIS Based on Vague Sets and its Application in the Scheme Evaluation of Mechanical Design

Fang Liu1, Guolei Zheng1, Weiping Wang2 and Daju Xu3

1. School of Mechanical Engineering and Automation, Beihang University, Beijing, 100191 E-mail: liu_f@me.buaa.edu.cn; zhengguolei@buaa.edu.cn

2. University of International Relations, Beijing, 100091, China E-mail: weipw@sina.com

3. Dept. of Mathematics and Physics, Shandong Jiaotong Univ., Jinan, 250023, China, E-mail: xudj@sdjtu.edu.cn

Abstract: This paper points out the same limitations of some main distance measures between vague sets at present. According to the basic rules of distance measures, a new distance formula is constructed based on the concept of which the influence of hesitancy/uncertain degree of vague values should be superimposed instead of being counteracted. This thesis also discussed the TOPSIS based on the distance measure, and its effectiveness in application fields which is illustrated by an example of scheme evaluation of mechanical design. It provides a new approach to solving decision making problems of scheme evaluation. Key Words: vague sets, TOPSIS, mechanical design, scheme evaluation

458978-1-4244-5182-1/10/$26.00 c©2010 IEEE

( ) ( ) ( ) ( ) ( )( )11

1,2

n

H A i B i A i B ii

D A B t x t x f x f xn =

= − + − (1)

( ) ( ) ( ) ( ) ( )2 21,1 2 1

nD A B t x t x f x f xE A i B i A i B in i

= − + −=

(2)

( ) ( ) ( ) ( ) ( )1, max ,2 2 1

nD A B t x t x f x f xi i i iB BH A An i

= − −=

(3)

( ) ( ) ( ) ( ) ( )2 21, max ,2 2 1

nD A B t x t x f x f xE A i B i A i B in i

= − −=

(4)

( ) ( ) ( ) ( ) ( ) ( ) ( )( )31

1,2

n

H A i B i A i B i A i B ii

D A B t x t x f x f x x xn

π π=

= − + − + − (5)

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

1

1,2

n

E A i B i A i B i A i B ii

D A B t x t x f x f x x xn

π π=

= − + − + − (6)

3 A NEW DISTANCE MEASURE BETWEEN VAGUE SETS

3.1 Analyses of Existing Distance Measures

Most existing distance measures discussed in the discrete field are mainly based on Minkowski metric or Hausdorff measure. According to the parameters associated with, the current distance measures are divided into two types. One adopts only two parameters- truth-membership functions and false-membership functions-, the other takes the uncertainty function into account excluding two membership functions. Atanassov introduced initial distance measures between intuitionistic fuzzy sets [10]. He presented a normalized Hamming distance 1HD and a Euclidean distance 1ED . Based on Hausdorff metric, Grzegorzewski [11] proposed a normalized Hamming distance 2HD and a Euclidean distance 2ED .Only truth-membership functions and false-membership functions are considered in formula (1) to (4), which will result in loss of information for ignoring uncertain information. Hence, some researchers attempted to evaluate the influence of uncertain information in the distance measures.Further discussion has been provided by Szmidt and Kacprzyk from the perspective of a geometrical representation of an intuitionistic fuzzy set [12]. They introduced a normalized Hamming distance 3HD and a Euclidean distance 3ED . These distance measures above are all correct and meets distance axioms from the point of view of pure mathematical conditions. However, some conclusions in accordance with these methods do not match our intuition, which will directly affect decision making results based on vague sets. A common ground of them is that

( ), 0D A B A B= ⇔ = .It is true in the universe of real number or fuzzy sets but it should be reconsidered in vague sets. E.g., the real situation in practice represented by [ ]0,1 might be any vague value [ ],1t f− .Two same

vague values [ ]0,1 might represent different cases in practice, perhaps they may represent two opposite situations [ ]0,0 and [ ]1,1 . Generally, people will not regard two uncertain conditions as one even if they are described in the same representation. A reasonable explanation might be “the distance between two same vague values (or vague sets) is zero if and only if there is no uncertain information within them”. That is to say, there should be some distance between two vague values (or vague sets) even they are the same as long as uncertain information exists. Furthermore, with the increase of uncertainty the distance will increase too.

3.2 Basic Rules of Distance Measures between Vague

Sets

Definition 3 suppose a mapping

( ) ( ) [ ]: VS VS 0,1D X X× → .

D is called a distance metric if for ( ), , VSA B C X∀ ∈ ,

D satisfies

(1) ( )0 , 1D A B≤ ≤ ;

(2) ( ) ( ), ,D A B D B A= ;

(3) ( ) ( ) ( ), , ,D A C D A B D B C≤ + ;

(4) ( ) ( ) ( ){ }, max , , , ,D A C D A B D B C≥

where , .A B C⊆ ⊆

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( ),D A B is called the distance between A and B .

(1) to (4) are the basic rules of distance measures between vague sets[13].

3.3 A New distance measure between Vague Sets

Definition 4 Suppose ( )RVVSA X∈ , for ,x y A∀ ∈ ,

[ ],1x xx t f= − , ,1y yy t f= − , the distance between

them is defined as

( ) ( ) ( )1 1,2 6x y x y x yD x y t t f f π π= − + − + + . (7)

It is easy to prove that ( , )D x y accords with the basic rules of distance measures .Besides, it has some properties bellow.

Theorem 1 (1) ( ) { } [ ] [ ]{ }, 1 , 0,0 , 1,1D x y x y= ⇔ = ;

(2) ( ), 0D x y x y= ⇔ = , and 0x yπ π= = ;

(3) Suppose [ ], , ,x x y yx t t y t t= = , then

( ), x yD x y t t= − ;

(4) ( ) ( ){ }, min , ,D x x D x y y A= ∀ ∈ ;

Proof properties (1) to (3) are obvious according to the definition ( , )D x y .

(4) Suppose [ ],1x xx t f= − , then ( ), 3π= xD x x .

For y∀ , ( ),1x xy t t f f= + Δ − + Δ ,

( ) ( ) ( ) ( )1 1, ,2 6

0

D x y D x x t f t f− = Δ + Δ − Δ + Δ

≥.

Hence, ( ) ( ){ }, min , ,D x x D x y y A= ∀ ∈ .

Remark Theorem 1 shows that the distance metric D can reflect our concept of distance measures between vague sets which takes uncertain information into account. Only when there is no uncertain information, can the distance between two vague values/sets be zero. At the same time, these vague values/sets reverse back to ordinary fuzzy values/sets. Especially, the distance between [0, 0] and [1, 1] is the only farthest distance. Although the distance between a vague value and itself is not equal to zero, but among all values the nearest element to a vague value is itself. So D is a generalization of distance definition in the fuzzy sets.

Definition 5 Let { }1 2, , nX x x x= be a universe,

( ), VSA B X∀ ∈ , ( ),D A B is a distance measure between A and B , where

( ),D A B = ( ) ( )1

1 1[ (2

n

A i B ii

t x t xn =

−

( ) ( ) ( ) ( )( )1) ]6A i B i A i B if x f x x xπ π+ − + +

Theorem 2 ( ), 0D A B A B= ⇔ = .

At the same time, for x A∀ ∈ , 0xπ = . It is obviously true according to Theorem 1, (2).

4 TOPSIS BASED ON THE DISTANCE BETWEEN VAGUE SETS

TOPSIS technique for order preference by similarity to ideal solution was proposed by Hwang and Yoon.[14] It uses PIS (Positive-Ideal Solution) and NIS (Negative-Ideal Solution) of a multiple criteria decision-making problem to rank programs. PIS is not a necessarily existing program, but is the best solution of virtual. NIS is a virtual worst program which does not necessarily exist too. The one that is most close to the PIS and most distant form NIS will be the best decision. So, we can judge the priority by calculating the distances between every program and PIS and NIS. Specific steps of TOPSIS based on the distance between vague sets are as follows. For a multi-criteria decision problem, suppose

1 2{ , , , }mA A A A= denotes the decision scheme set

and 1 2{ , , , }nC C C C= denotes the attribute set. The attribute values of decision schemes are given by vague values as follow:

iA = ( ) ( ) ( ){ }1 1 2 2, , , , , ,i i n inc v c v c v= , where

,1ij ij ijv t f= − , 1,2, , .j n=

Step 1 Construct PIS +A and NIS −A .

A+ ( ) ( ) ( ){ }1 1 2 2, , , , , ,n nc v c v c v+ + += ,

A− ( ) ( ) ( ){ }1 1 2 2, , , , , ,n nc v c v c v− − −= .

ijv+ and ijv

− are determined according to the property of the

attribute jc . If jc is a attribute of cost type,

then [0,0]jv+ = , [1,1]jv

− = . If jc is a attribute of benefit

type, then [0,0]jv− = , [1,1]jv

+ = .

Step 2 Calculate the distances between each scheme to

PIS A+ and NIS −A .

Assume { }nω ω ω= 1 2, , , is the weights set,

where iω is the weight of ix , and1

1n

ii

ω=

= .Using the new

distance measure proposed in section 2, according to formula (7), the distance between iA to PIS A+ is defined as

( , )i iD D A A+ += ( )1

, ;n

j ij jj

D v vω +

=

= (8)

460 2010 Chinese Control and Decision Conference

Table Attribute values of each scheme

Scheme 1x 2x 3x * 4x 5x * 6x * 7x *

A1 [0.83,0.83] [0.83,0.83] [0.8,0.9] [0.79,0.79] [0.4,0.6] [0.6,0.7] [0.7,0.8]

A2 [0.75,0.75] [0.67,0.67] [0.85,0.9] [0.96,0.96] [0.8,0.9] [0.8,0.9] [0.8,0.85]

A3 [1,1] [1,1] [0.6,0.8] [0.83,0.83] [0.2,0.3] [0.2,0.4] [0.65,0.75]

A4 [0.92,0.92] [0.92,0.92] [0.65,0.75] [1,1] [0.6,0.7] [0.7,0.8] [0.9,0.95]

A+ [0,0] [0,0] [1,1] [0,0] [1,1] [1,1] [1,1]

A− [1,1] [1,1] [0,0] [1,1] [0,0] [0,0] [0,0]

Table each scheme’s evaluation value and rank

Scheme iD+

iD−

iR Rank

A1 0.683 0.328 0.324 1

A2 0.705 0.301 0.299 2

A3 0.788 0.223 0.221 3

A4 0.786 0.221 0.219 4

The distance between iA to NIS A− is defined as

( , )i iD D A A− −= ( )1

,n

j ij jj

D v vω −

=

= (9)

Step 3 calculate each scheme’s evaluation value iR , where

ii

i i

DRD D

−

− +=+

, 1,2, ,i m= (10)

Step 4 rank schemes according to the evaluation values. The larger iR is, the better iA is.

5 THE APPLICATION OF TOPSISS IN THE SCHEME EVALUATION OF MECHANICAL DESIGN

Generally, in mechanical design, there are multiple design indexes which are usually restricting each other. Different focus on design goals can produce different schemes. In order to obtain a plan that is feasible in technique advanced in performance, rational in economy and of higher reliability, it is necessary to evaluate schemes. This is not only the scientific analysis and summary of schemes, but also an optimization process in the development of products. For instance, an electric machine design includes multiple design indexes: Manufacturing cost and power consumption, manufacturing and maintenance convenience, reliability, the total weight of the machine and so on. Designers often have to take various factors into account and choose a satisfactory solution. Because the characteristic of attributes are clear, the features of optimal

scheme and the worst are definite. Therefore TOPPSIS method can be used to select the optimal solutions. When designing a 1000kN electric machine[9], according to the experts’ analysis and design experience, manufacturing cost ( 1x ), machine total weight ( 2x ),

structure rationality ( 3x ), power consumption ( 4x ), the

difficulty of manufacturing ( 5x ), maintenance convenience

( 6x ) and reliability ( 7x ) are chosen main indexes (attributes). The attribute set is

1 2 3 4 5 6 7 8{ , , , , , , , }X x x x x x x x x= .

And each attribute values are given by vague values. Suppose decision makers focus on manufacturing cost, power consumption and reliability, and think others are secondary. Assume the attribute weights are

( 0. 2, 0.05, 0. 05, 0. 5, 0. 05, 0. 05, 0. 1)=We should select the optimal scheme among scheme A1-A4.In table 1, the attributes with * are qualitative evaluation indexes given by expert. Manufacturing cost ( 1x ), machine total weight ( 2x ), power consumption ( 4x )are cost type attributes, which means “the smaller the better”. And others are benefit type, which means “the larger the better”. So, we can construct PIS +A and NIS −A in table .Using TOPSISS proposed in this paper, according to formula (8) and (9), the distance between iA to A+ and

A− are calculated. We can obtain each scheme’s

2010 Chinese Control and Decision Conference 461

evaluation value to rank the order (Table 2). It is easy to see that 1 2 3 4A A A A . So, 1A is the optimal scheme.

6 Conclusion Using vague sets can reflect realities more objectively. It conforms to the people's understanding and objective reality. Therefore, its application scope would be extensive and practical. In this paper, TOPPSIS based on vague sets was proposed. This method can rank schemes according to decision-makers’ expectation of attributes. The validity of this method is illustrated with an example in mechanical design.

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[8] W.Q. Wang, X.L. Xin. Distance measure between intuitionistic fuzzy sets, Pattern Recognition Letters 26 , 2063–2069, 2005.

[9] Ye Jun, Lou Jian-guo, Li Wei-bo, Decisions-making of machine design schemes based on similarity Measures between vague sets, Mechanical science and technology, 2005, Vol. 24, No. 10, 1173-1175.

[10] Atanassov K., Intuitionistic fuzzy sets theory and applications, Heidelberg, New York: Physica-verl., 1999.

[11] Przemyslaw G., Distances between intuitionistic fuzzy sets and/or interval valued fuzzy sets based on the Hausdorff metric, Fuzzy Sets and Systems, 148, 319–328, 2004.

[12] Szmidt, E., Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets Systems, 114 (3), 505–518, 2000.

[13] Weiping Wang, Qi-zong Wu, Yu Shi, Similarity Measures between Interval Value Vague Sets Based on Distance, Proceedings of 2008 Chinese Control and Decision conference, 2008CCDC, 2300-2305, Yantai, China.

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