5441336550

problem. Also an example is presented.

2 Preliminaries

Also we write a < b if and only if b > a\ a < b if R    R R

and only if b > a.

R

2.1 Vector ranking function

Let x = (xi,X2, ■ ■ ■ ,x_n)', y = (2/1,2/2, • • -, y_n)' e R be two vectors, where ”/” denotes transpose of the vector. Then we write x > y if and only if Xi>yi, for all i belong to N = {1,2,..., n}; x > y if and only if Xi > yi, for all * G N; x y if and only if Xi ^ y%, for some i £ N.

Definition 2.1 [3] A fuzzy set a on IR is called a fuzzy nurnber if it holds:

1)    Its membership function is upper semi contin-uous.

2)    There exist three interual [a, 6], [6, c], [c, d] such that a is increasing on [a, 6] , equal to 1 on [6, c] ,decreasing on [c, d\ and eąual to 0 anywhere else.

We denote the set of all fuzzy numbers by F( R ). A simple method for ordering the el-ements of F( IR ) consists in the defining of a ranking function R : F( R ) —» IR which maps each fuzzy number into a real number, where a natural order exists. It is obvious that morę than one ranking function can be defined [2,3].

Based on the decision maker’s Preferences, as-sume there exist k important attributes associ-ated to fuzzy number a such that the th of them can be characterized by the ranking function Ri : F( R ) —» R . In this case, we asso-ciate a crisp fc-dimensional vector, R(a), to a as follows:

R(a) = (Ri(a), R2(a),..., Rk(d))'.

Definition 2.2 The uector function R(.), defined as aboue, is called a uector of ranking func-tions. Moreouer, let a and b belong to F( R ), then :

•    a > b if and only if R(a) > R(6).

R

•    a > b if and only if R(a) > R(6).

R

•    a = b if and only if R(a) = R(6).

•    a ^ b if and only if R(a) 7^ R(6).

R

Example 2.3 Let a be a fuzzy number.

a) For k = 1, we consider the Roubens ranking function [1 ] which is defined as:

R(Or)

= 1/2 f (infa_T Jo

- supa_r)dr,

where a_r is an r-cut of a (0 < r < 1) i.e, a_r = {x G R |a(x) > r}.

b) For k = 2, consider

R(a) = (E(a),—Var(a))',

where E(a) and Var(a) are the expectation and uariance of the density function associated uńth a. See [6].

c) For k = 3, consider

R(a) = (V(a),A(d),F(a))'

where V(a), A(a) and F(a) are ualue, ambiguity and fuzziness ofa, respectiuely, which are defined as:

V(a) = [ r[Lg,(r) + Ra(r)]dr,

Jo

A(a)= frlMO-L^dr,

Jo

rl/2

F(a)= [Jfc(r) - L_s(r)]dr Jo

+ f [La(r)-Ra(r)]dr,

J1/2

where La(.) and i?a(-) both from [0,1] to R defined by

isW

{ii ii

inf{x\x e a_r} inf{x\x e Supp{a)}

sup{x\x e Or} sup{x\x e Supp(a)} See [2], [3].

Ra(r)

ifr G (0,1], ifr = 0.

ifr G (0,1], ifr — 0.